The document analyzes whether it is feasible for a student to throw a rugby ball over the roof of their house, run through the house, and catch the ball as it comes down in the garden below. Using assumptions about house dimensions, throwing and running speeds, the document creates a mathematical model to analyze an optimal scenario. It determines that with a throw angle of 81.2 degrees, the time the ball is in the air is 3.61 seconds, which is longer than the 1.392 seconds needed for the student to run through the house. Therefore, the scenario is deemed feasible in an optimal setting.
The document summarizes an investigation into Hooke's law. Students will apply increasing forces to a spring by adding masses and measure the corresponding extension. They will record their results in a table and graph, and analyze the relationship between force and extension to discover Hooke's law - that a spring's extension is proportional to the applied force, up to the elastic limit where the spring may break. On Friday, the class will have an end of topic test covering all material learned that year.
The law of the lever states that the product of the effort force and its distance from the fulcrum is equal to the product of the load force and its distance from the fulcrum. This relationship is expressed as F x d1 = R x d2, where F is the effort force, R is the load force, d1 is the distance from the fulcrum to the effort, and d2 is the distance from the fulcrum to the load. The lever is in equilibrium when this equation is satisfied.
Hooke's law sample problems with solutionsBasic Physics
A graph of force (F) versus elongation (x) shown in figure below. Find the spring constant!
Solution
Hooke's law formula :
k = F / x
F = force (Newton)
k = spring constant (Newton/meter)
x = the change in length (meter)
The document contains multiple physics problems related to rotation, torque, angular acceleration, and moments of inertia. One problem involves calculating the angular speed of a winch drum that is lifting a 2000 kg block at a constant speed of 8 cm/s. It is determined that:
- The tension in the cable equals 19.6 kN, given by the weight of the block
- The torque exerted on the winch drum by the cable is 5.9 mkN
- The angular speed of the winch drum is 0.27 rad/s
- The power required by the motor is 1.6 kW
1. The document analyzes the kinematics of a ball thrown upward with an initial velocity (Vi) of 15 m/s, experiencing a constant downward acceleration (a) of 9.8 m/s^2.
2. It is calculated that the ball reaches its maximum height of 11.479 meters after 1.5306 seconds, at which point its velocity is 0 m/s.
3. If not caught, the ball will then fall freely back to the ground, following the kinematic equations of constant downward acceleration and reaching higher velocities as it falls.
The document analyzes the failure of a bike pedal spindle. It determines the location of fatigue crack origin and calculates various forces, moments, and stresses acting on the spindle. The largest stress was found to be torque stress. This agrees with observations that the spindle failed in a perpendicular direction to the torque stress. Based on the images and fracture mechanics analysis, the critical crack length and size of the plastic zone were estimated. The maximum pedal force required to cause final fracture was also calculated based on the material's fracture toughness.
1) A student pointed out that a sailboat with a fan blowing into its sails could in fact move forward, because the air molecules bouncing off the sail would impart twice the momentum change experienced going through the fan, providing a net forward force.
2) A 2000-kg car traveling at 30 m/s collided perfectly inelastically with another 2000-kg car traveling at 10 m/s, sticking together with a final speed of 20 m/s. 20% of the initial kinetic energy was lost to heat and deformation.
3) A 16-g bullet fired into a 1.5-kg ballistic pendulum was calculated to have a speed of 45 m/s before impact by applying
This document discusses vertical projectile motion with an initial vertical velocity component. It explains that when an object is projected at an angle between 0 and 90 degrees, it will have both horizontal and vertical velocity components. The vertical component is given by the initial velocity times the sine of the angle. It provides examples of calculating time of flight, maximum height, and range for objects projected at an angle. It also notes that the greatest range occurs at 45 degrees and greatest time of flight at 90 degrees.
The document summarizes an investigation into Hooke's law. Students will apply increasing forces to a spring by adding masses and measure the corresponding extension. They will record their results in a table and graph, and analyze the relationship between force and extension to discover Hooke's law - that a spring's extension is proportional to the applied force, up to the elastic limit where the spring may break. On Friday, the class will have an end of topic test covering all material learned that year.
The law of the lever states that the product of the effort force and its distance from the fulcrum is equal to the product of the load force and its distance from the fulcrum. This relationship is expressed as F x d1 = R x d2, where F is the effort force, R is the load force, d1 is the distance from the fulcrum to the effort, and d2 is the distance from the fulcrum to the load. The lever is in equilibrium when this equation is satisfied.
Hooke's law sample problems with solutionsBasic Physics
A graph of force (F) versus elongation (x) shown in figure below. Find the spring constant!
Solution
Hooke's law formula :
k = F / x
F = force (Newton)
k = spring constant (Newton/meter)
x = the change in length (meter)
The document contains multiple physics problems related to rotation, torque, angular acceleration, and moments of inertia. One problem involves calculating the angular speed of a winch drum that is lifting a 2000 kg block at a constant speed of 8 cm/s. It is determined that:
- The tension in the cable equals 19.6 kN, given by the weight of the block
- The torque exerted on the winch drum by the cable is 5.9 mkN
- The angular speed of the winch drum is 0.27 rad/s
- The power required by the motor is 1.6 kW
1. The document analyzes the kinematics of a ball thrown upward with an initial velocity (Vi) of 15 m/s, experiencing a constant downward acceleration (a) of 9.8 m/s^2.
2. It is calculated that the ball reaches its maximum height of 11.479 meters after 1.5306 seconds, at which point its velocity is 0 m/s.
3. If not caught, the ball will then fall freely back to the ground, following the kinematic equations of constant downward acceleration and reaching higher velocities as it falls.
The document analyzes the failure of a bike pedal spindle. It determines the location of fatigue crack origin and calculates various forces, moments, and stresses acting on the spindle. The largest stress was found to be torque stress. This agrees with observations that the spindle failed in a perpendicular direction to the torque stress. Based on the images and fracture mechanics analysis, the critical crack length and size of the plastic zone were estimated. The maximum pedal force required to cause final fracture was also calculated based on the material's fracture toughness.
1) A student pointed out that a sailboat with a fan blowing into its sails could in fact move forward, because the air molecules bouncing off the sail would impart twice the momentum change experienced going through the fan, providing a net forward force.
2) A 2000-kg car traveling at 30 m/s collided perfectly inelastically with another 2000-kg car traveling at 10 m/s, sticking together with a final speed of 20 m/s. 20% of the initial kinetic energy was lost to heat and deformation.
3) A 16-g bullet fired into a 1.5-kg ballistic pendulum was calculated to have a speed of 45 m/s before impact by applying
This document discusses vertical projectile motion with an initial vertical velocity component. It explains that when an object is projected at an angle between 0 and 90 degrees, it will have both horizontal and vertical velocity components. The vertical component is given by the initial velocity times the sine of the angle. It provides examples of calculating time of flight, maximum height, and range for objects projected at an angle. It also notes that the greatest range occurs at 45 degrees and greatest time of flight at 90 degrees.
The document provides a review of units and the metric system. It discusses the advantages of the metric system over the US customary system, including its use of consistent prefixes that are multiples of 10. It also covers converting between units, scientific notation, combined units like those for speed and temperature, and basic algebra review. The document aims to prepare students for material covered in an upcoming science course through this units and math review.
This document discusses Hooke's law of elasticity and provides examples of its application. [1] Hooke's law states that the extension of a spring is proportional to the applied load as long as the elastic limit is not exceeded. [2] Materials that obey this law are known as linear-elastic or "Hookean" materials. [3] The document then provides the formula for Hooke's law, F=kx, and gives two examples of using it to calculate spring force and spring constant.
Wedges are simple machines that transform applied forces into larger forces directed at approximately right angles. They are used to give small adjustments to heavy loads by applying a smaller force. To analyze wedges, force and moment equilibrium equations are used along with frictional force equations, with unknown normal and frictional forces. Wedges can be self-locking if the frictional forces alone are enough to hold the load without any applied force needed.
- A block rests on an inclined plane making an angle of 30° with the horizontal.
- Static friction acts on the block to prevent it from sliding down the plane.
- The coefficient of static friction (μs) between the block and plane can be expressed as μs = tanθ.
- Since the plane makes an angle of 30° with the horizontal, the coefficient of static friction is equal to tan30° = 0.577.
The document discusses Hooke's law and moments in physics. It explains that Hooke's law states that the extension of an object is proportional to the applied load as long as the elastic limit is not exceeded. It also defines the terms elasticity, plasticity, extension, and elastic limit. Additionally, it covers the turning effect of forces, or moments, and defines key terms related to moments like pivot, moment equation, and principle of moments.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
1. Torque is the rotational equivalent of force and produces angular acceleration in an object. It is a vector quantity equal to the cross product of the position vector and applied force.
2. Torque depends on the magnitude of the applied force and the moment arm, which is the perpendicular distance between the axis of rotation and the line of action of the force.
3. Collisions can be elastic, where both momentum and kinetic energy are conserved, or inelastic, where only momentum is conserved but not kinetic energy.
This experiment investigates Hooke's Law, which states that the force (F) needed to extend or compress a spring by some distance (Δx) is proportional to that distance. Specifically, F = kΔx, where k is a constant called the spring constant. The experiment involves measuring the position (x) of a spring when hung with different masses (m). This allows the calculation of the force due to gravity (Fg) and the stretching distance (Δx). Plotting Fg versus Δx should produce a straight line, verifying the proportional relationship between force and distance. The slope of the line gives the value of k, the spring constant. A second method will also be used to independently determine k to
The document discusses work, kinetic energy, and power as they relate to mechanical systems. It includes conceptual problems with explanations of the relevant physics concepts and mathematical solutions. Specifically:
- Problem 7 compares the work required to stretch a spring different distances based on the equation that work done on a spring is proportional to the square of its displacement.
- Problem 13 tests understanding of scalar products by asking whether several statements about scalar products and vectors are true or false.
- Problem 17 explains that the only external force doing work to accelerate a car from rest is friction between the tires and the road, using free body diagrams and the work-kinetic energy theorem.
This document discusses periodic motion in a simple pendulum. It provides the details to solve two problems:
(1) Find the displacement and acceleration of a 5kg pendulum with a 2m string displaced at an angle of 45 degrees. The displacement is calculated to be 1.57m and the acceleration is 6.93m/s^2.
(2) Assuming a very small angle of displacement, the angular frequency is calculated to be 2.21rad/s. Reducing the angle of displacement would decrease the acceleration, as the acceleration equation is directly proportional to the displacement angle.
The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.
The document discusses moments, which are the tendency of a force to cause rotation about an axis. It defines key terms like moment, moment arm, and how to calculate moment using the equation: Moment = Force x Perpendicular Distance. It also covers units of moment, properties like sense of direction, and applications like Varignon's theorem for resolving forces. An example problem is worked through to find the moment created by different forces and placements.
The document describes the mass, dimensions, and notation used in a problem involving the forces exerted on an automobile's wheels. It provides the mass M of the automobile, horizontal distances L between axles and from the rear axle to the center of mass, and defines the forces F1 and F2 exerted on the front and back wheels, respectively. Taking torques about the rear axle allows calculating F1 as 2.77 × 103 N, and equilibrium of forces gives an equation to solve for F2 as 3.89 × 103 N.
1) The document provides instructions for students to work through a physics problem involving calculating the forces acting on Spiderman suspended from two webs.
2) The problem involves drawing force diagrams, using trigonometry to find components of the tension forces, and applying Newton's Second Law to calculate the tension in one of the webs.
3) Working through the steps provided, the tension in the second web (T2) is calculated to be approximately 636N or 637N.
- Shear stress distribution in beams takes a parabolic shape, with the maximum stress at the neutral axis and zero at the ends. In rectangular beams the stress is highest at y=0. In I-beams, most stress is carried by the web in a "top-hat" distribution.
- Circular beams have a shear stress distribution that also follows a parabolic shape, calculated using the area moment of the shaded portion.
- Principal stresses can be determined in beams using the bending and shear stresses. The bending stress is not a principal stress and the principal stresses are found using an equation involving the bending and shear stresses.
This document provides steps to calculate the β value for a half-wave rectifier circuit with an R-L load. It is given that Vs=240V, f=50Hz, R=100Ω, and L=0.1H. The document calculates the θ and ωτ values, then sets up and solves a function to find the value of β that satisfies the equation. Through iterative calculations using this function, it determines that the β value is approximately 3.4.
Matrix metalloproteinases (MMPs) are enzymes that degrade components of the extracellular matrix. They are implicated in various neurological conditions and play important roles in processes like synaptic remodeling. More recently, MMPs have been shown to regulate neural stem cell biology and remyelination, suggesting their importance in nervous system regeneration. While MMPs serve beneficial functions, their upregulation in conditions like neuroinflammation and neurodegeneration can also contribute to tissue damage.
MMP's and the Role of Zinc in Wound Healingmvkaminski
Matrix metalloproteinase (MMP) and Zinc play an important role in wound healing. As MMP's are dependent upon Zinc, you can't have one without the other. Without MMP's you can not have collenganese activity (or other endogenous proteolytic enzymes) resulting in a wound that will not heal. For more information on this topic and many others regarding wound healing, check out our other presentations here.
Matrix metalloproteinases (MMPs) are a family of calcium-dependent zinc-containing endopeptidases that are responsible for tissue remodeling and degradation of the extracellular matrix. MMPs are excreted by connective tissue and inflammatory cells and play a role in both physiological and pathological processes such as angiogenesis. MMPs degrade various components of the extracellular matrix, including collagens, and their activation leads to tissue remodeling and degradation involved in conditions like cancer.
The role of matrix metalloproteinase 2 Feng-wei Yeh
This document discusses the role of matrix metalloproteinase 2 (MMP-2) in chronic periodontitis. MMP-2 is a gelatinase found in gingival crevicular fluid and periodontal tissue that can degrade proteins and collagen fibers. Increased levels of MMP-2 may be associated with the symptoms of chronic periodontitis such as inflammation, tissue destruction, and tooth mobility. Quantifying MMP-2 levels in gingival crevicular fluid could help diagnose chronic periodontitis and monitor the effects of treatment. Natural compounds that inhibit MMP-2 may offer alternative therapies to antibiotics for treating chronic periodontitis.
The document provides a review of units and the metric system. It discusses the advantages of the metric system over the US customary system, including its use of consistent prefixes that are multiples of 10. It also covers converting between units, scientific notation, combined units like those for speed and temperature, and basic algebra review. The document aims to prepare students for material covered in an upcoming science course through this units and math review.
This document discusses Hooke's law of elasticity and provides examples of its application. [1] Hooke's law states that the extension of a spring is proportional to the applied load as long as the elastic limit is not exceeded. [2] Materials that obey this law are known as linear-elastic or "Hookean" materials. [3] The document then provides the formula for Hooke's law, F=kx, and gives two examples of using it to calculate spring force and spring constant.
Wedges are simple machines that transform applied forces into larger forces directed at approximately right angles. They are used to give small adjustments to heavy loads by applying a smaller force. To analyze wedges, force and moment equilibrium equations are used along with frictional force equations, with unknown normal and frictional forces. Wedges can be self-locking if the frictional forces alone are enough to hold the load without any applied force needed.
- A block rests on an inclined plane making an angle of 30° with the horizontal.
- Static friction acts on the block to prevent it from sliding down the plane.
- The coefficient of static friction (μs) between the block and plane can be expressed as μs = tanθ.
- Since the plane makes an angle of 30° with the horizontal, the coefficient of static friction is equal to tan30° = 0.577.
The document discusses Hooke's law and moments in physics. It explains that Hooke's law states that the extension of an object is proportional to the applied load as long as the elastic limit is not exceeded. It also defines the terms elasticity, plasticity, extension, and elastic limit. Additionally, it covers the turning effect of forces, or moments, and defines key terms related to moments like pivot, moment equation, and principle of moments.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
1. Torque is the rotational equivalent of force and produces angular acceleration in an object. It is a vector quantity equal to the cross product of the position vector and applied force.
2. Torque depends on the magnitude of the applied force and the moment arm, which is the perpendicular distance between the axis of rotation and the line of action of the force.
3. Collisions can be elastic, where both momentum and kinetic energy are conserved, or inelastic, where only momentum is conserved but not kinetic energy.
This experiment investigates Hooke's Law, which states that the force (F) needed to extend or compress a spring by some distance (Δx) is proportional to that distance. Specifically, F = kΔx, where k is a constant called the spring constant. The experiment involves measuring the position (x) of a spring when hung with different masses (m). This allows the calculation of the force due to gravity (Fg) and the stretching distance (Δx). Plotting Fg versus Δx should produce a straight line, verifying the proportional relationship between force and distance. The slope of the line gives the value of k, the spring constant. A second method will also be used to independently determine k to
The document discusses work, kinetic energy, and power as they relate to mechanical systems. It includes conceptual problems with explanations of the relevant physics concepts and mathematical solutions. Specifically:
- Problem 7 compares the work required to stretch a spring different distances based on the equation that work done on a spring is proportional to the square of its displacement.
- Problem 13 tests understanding of scalar products by asking whether several statements about scalar products and vectors are true or false.
- Problem 17 explains that the only external force doing work to accelerate a car from rest is friction between the tires and the road, using free body diagrams and the work-kinetic energy theorem.
This document discusses periodic motion in a simple pendulum. It provides the details to solve two problems:
(1) Find the displacement and acceleration of a 5kg pendulum with a 2m string displaced at an angle of 45 degrees. The displacement is calculated to be 1.57m and the acceleration is 6.93m/s^2.
(2) Assuming a very small angle of displacement, the angular frequency is calculated to be 2.21rad/s. Reducing the angle of displacement would decrease the acceleration, as the acceleration equation is directly proportional to the displacement angle.
The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.
The document discusses moments, which are the tendency of a force to cause rotation about an axis. It defines key terms like moment, moment arm, and how to calculate moment using the equation: Moment = Force x Perpendicular Distance. It also covers units of moment, properties like sense of direction, and applications like Varignon's theorem for resolving forces. An example problem is worked through to find the moment created by different forces and placements.
The document describes the mass, dimensions, and notation used in a problem involving the forces exerted on an automobile's wheels. It provides the mass M of the automobile, horizontal distances L between axles and from the rear axle to the center of mass, and defines the forces F1 and F2 exerted on the front and back wheels, respectively. Taking torques about the rear axle allows calculating F1 as 2.77 × 103 N, and equilibrium of forces gives an equation to solve for F2 as 3.89 × 103 N.
1) The document provides instructions for students to work through a physics problem involving calculating the forces acting on Spiderman suspended from two webs.
2) The problem involves drawing force diagrams, using trigonometry to find components of the tension forces, and applying Newton's Second Law to calculate the tension in one of the webs.
3) Working through the steps provided, the tension in the second web (T2) is calculated to be approximately 636N or 637N.
- Shear stress distribution in beams takes a parabolic shape, with the maximum stress at the neutral axis and zero at the ends. In rectangular beams the stress is highest at y=0. In I-beams, most stress is carried by the web in a "top-hat" distribution.
- Circular beams have a shear stress distribution that also follows a parabolic shape, calculated using the area moment of the shaded portion.
- Principal stresses can be determined in beams using the bending and shear stresses. The bending stress is not a principal stress and the principal stresses are found using an equation involving the bending and shear stresses.
This document provides steps to calculate the β value for a half-wave rectifier circuit with an R-L load. It is given that Vs=240V, f=50Hz, R=100Ω, and L=0.1H. The document calculates the θ and ωτ values, then sets up and solves a function to find the value of β that satisfies the equation. Through iterative calculations using this function, it determines that the β value is approximately 3.4.
Matrix metalloproteinases (MMPs) are enzymes that degrade components of the extracellular matrix. They are implicated in various neurological conditions and play important roles in processes like synaptic remodeling. More recently, MMPs have been shown to regulate neural stem cell biology and remyelination, suggesting their importance in nervous system regeneration. While MMPs serve beneficial functions, their upregulation in conditions like neuroinflammation and neurodegeneration can also contribute to tissue damage.
MMP's and the Role of Zinc in Wound Healingmvkaminski
Matrix metalloproteinase (MMP) and Zinc play an important role in wound healing. As MMP's are dependent upon Zinc, you can't have one without the other. Without MMP's you can not have collenganese activity (or other endogenous proteolytic enzymes) resulting in a wound that will not heal. For more information on this topic and many others regarding wound healing, check out our other presentations here.
Matrix metalloproteinases (MMPs) are a family of calcium-dependent zinc-containing endopeptidases that are responsible for tissue remodeling and degradation of the extracellular matrix. MMPs are excreted by connective tissue and inflammatory cells and play a role in both physiological and pathological processes such as angiogenesis. MMPs degrade various components of the extracellular matrix, including collagens, and their activation leads to tissue remodeling and degradation involved in conditions like cancer.
The role of matrix metalloproteinase 2 Feng-wei Yeh
This document discusses the role of matrix metalloproteinase 2 (MMP-2) in chronic periodontitis. MMP-2 is a gelatinase found in gingival crevicular fluid and periodontal tissue that can degrade proteins and collagen fibers. Increased levels of MMP-2 may be associated with the symptoms of chronic periodontitis such as inflammation, tissue destruction, and tooth mobility. Quantifying MMP-2 levels in gingival crevicular fluid could help diagnose chronic periodontitis and monitor the effects of treatment. Natural compounds that inhibit MMP-2 may offer alternative therapies to antibiotics for treating chronic periodontitis.
Alzheimer's disease is a neurological disorder that causes memory loss and cognitive decline. It was first described by Dr. Alois Alzheimer in 1901 when examining patient Auguste D. The causes are not fully understood but involve genetic, environmental, and lifestyle factors. Early signs include memory problems and other cognitive declines. As it progresses, damage occurs in areas controlling language, reasoning, and thought, and patients have trouble recognizing family and friends. By the final stage, plaques and tangles have spread throughout the brain, causing severe impairment and dependence on others for care. While some drugs can temporarily stabilize symptoms, there is no cure for Alzheimer's disease.
Alzheimer's disease is a progressive degenerative disorder of the brain that initially involves memory loss and cognitive decline, and ultimately results in severe impairment in all areas of functioning. While medications can temporarily improve symptoms, there is no cure. The disease progresses through mild, moderate, and severe stages characterized by worsening memory loss, impaired communication and ability to care for oneself, and may eventually involve inability to walk or speak intelligibly. Patients and families require education and support to understand and cope with the progression of the disease.
Alzheimer's disease is a progressive brain disorder that destroys memory and cognitive skills. Dr. Alois Alzheimer first described it in 1906 after examining a woman with dementia. The disease is characterized by beta-amyloid plaques and neurofibrillary tangles in the brain. Current treatments aim to improve symptoms but do not stop the underlying disease process. Researchers are exploring therapies targeting amyloid and tau proteins as well as other mechanisms to find a cure.
Alzheimer's disease is a progressive brain disease that causes memory loss and cognitive decline. It is the most common cause of dementia among older adults. The disease is characterized by two hallmarks - neuritic plaques formed by amyloid-beta protein fragments, and neurofibrillary tangles made up of tau protein inside neurons. It gradually destroys brain cells in areas responsible for memory and cognition. While symptoms start out mild, the disease gets worse over time and can lead to severe brain damage. There are genetic and lifestyle risk factors associated with Alzheimer's but currently there is no cure.
The role of matrix metalloproteinase 2 (mmp 2Feng-wei Yeh
The document discusses the role of matrix metalloproteinase 2 (MMP-2) in chronic periodontitis. MMP-2 degrades extracellular matrix proteins and collagen fibers. Due to its gelatinase function, MMP-2 may be associated with tooth mobility caused by periodontitis. Smoking is a major risk factor for chronic periodontitis by worsening inflammation. Tetracycline and doxycycline can inhibit MMPs and reduce periodontal inflammation by functioning as tissue inhibitors of metalloproteinases.
The document contains sample questions on various topics like basics of C programming, physics, trigonometry, probability, electric circuits, ratios, velocity, time and distance problems. It also includes questions related to permutation, area of shapes, progression of numbers, angular velocity and vampires. The questions are aimed at preparing an aptitude question bank covering different concepts.
The document describes an experiment conducted by Justin Park, Sean Kim, Paul Wang, and Derek Guo to measure the acceleration due to gravity (g) using a tennis ball dropped from various heights of a stairway. They recorded the height and time it took for the ball to hit the ground from each drop and calculated t-squared. By plotting displacement vs t-squared, they calculated the slope which equals g and found it to be 9.02 m/s2, close to the accepted value of 9.81 m/s2. Limitations included short time intervals and potential measurement errors. The conclusion is that the experiment provided a reasonably accurate value of g.
This document discusses several topics related to calculus including:
1) Derivatives of position, velocity, and acceleration and how they relate to each other.
2) An example problem calculating velocity from a position function.
3) The Mean Value Theorem and how to apply it to find critical points of a function.
4) How the first and second derivatives of a function relate to critical points, maxima, minima, and points of inflection or concavity.
5) Related rates problems and how to set them up using derivatives and relationships between variables.
College Physics 1st Edition Etkina Solutions ManualHowardRichsa
This document contains a chapter from the textbook "College Physics" by Etkina, Gentile, and Van Heuvelen. It includes multiple choice and conceptual questions about kinematics concepts like displacement, velocity, acceleration. It also includes practice problems asking students to draw motion diagrams and choose reference frames. The key concepts covered are scalar and vector quantities, relationships between displacement, velocity and acceleration, and using graphs to represent motion.
This document discusses motion with constant acceleration. It describes two cases of motion with positive acceleration: 1) an object moving away from an initial point P0 with increasing velocity, and 2) an object initially moving away from P0 with decreasing velocity until stopping and changing direction. Examples are provided of vertical motion under gravitational acceleration, including free fall, projectile motion upward and downward. The use of calculus integrals to derive kinematic equations from known acceleration is also discussed.
This document discusses rectilinear motion under gravity. It defines motion under gravity as motion of a particle projected vertically in air under the influence of gravitational force. It presents the equations of rectilinear motion and modifies them for motion under gravity. It notes that time of flight is the total time an object remains in air, which is the sum of time for upward and downward journeys. It provides examples calculating the time and velocity at which two objects crossing paths under gravity conditions.
The document explains how to use the Pythagorean theorem to solve problems involving right triangles. It provides examples of using the theorem to calculate the length of a ladder/slide, the distance a diver must swim, the length of a kite string, and the distance traveled from driving directions. The theorem states that for any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. The document walks through setting up and solving examples using the formula a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.
This document provides an introduction to linear kinematics. It discusses key linear kinematic variables like distance, displacement, speed, velocity, and acceleration. It defines these variables and the units used to measure them. It also describes the difference between scalar and vector quantities as they relate to motion. Examples of single-point and multi-segment models for describing motion are provided. Equations for calculating speed, velocity, and acceleration from changes in distance, displacement, and time are shown. Projectile motion is also summarized, including the independent vertical and horizontal components of projectile motion.
This document discusses projectile motion and provides examples. It can be analyzed as independent horizontal and vertical motion components. Projectile paths can be modeled using hanging beads on a ruler. Example problems calculate the time and speed of a ball rolling off a table and the maximum speed of a tennis ball clearing a net within the court boundaries, showing that mass does not affect projectile motion.
The document discusses motion in two and three dimensions. It defines key concepts like displacement, velocity, acceleration, and their representations as vectors. It also covers projectile motion, describing how the horizontal and vertical components are independent, and how to calculate range, time of flight and maximum height. Other topics covered include circular motion, uniform circular motion, relative motion using an intermediate frame of reference, and examples/activities calculating values for motion scenarios.
1. Two spacecraft, S and S', are moving relative to each other at a constant velocity v. S' measures a distance x' between itself and a light flash, and time t' on its clock, while S measures a distance x and time t.
2. Due to their relative motion, S and S' assign errors to each other's measurements, which can be accounted for by the Lorentz factor γ. Applying this leads to transformations between x and x', and t and t'.
3. These transformations imply that moving clocks run slow (time dilation) and moving objects appear contracted (length contraction) relative to an observer, in accordance with Einstein's theory of relativity.
A projectile is any object that moves freely through space under the influence of gravity. It follows a parabolic trajectory. Projectile motion is described using equations of linear motion as the projectile moves simultaneously in horizontal and vertical directions. The initial velocity can be resolved into horizontal and vertical components. The maximum height and range of a projectile depend on the initial velocity and angle of projection. Equations are derived to calculate velocities, displacements, flight time, maximum height and range for projectiles projected on horizontal and elevated surfaces. Examples show applications to problems involving projectile motion.
1) Gravitational acceleration is the acceleration experienced by objects due to gravity in the absence of other forces like air resistance. On Earth, gravitational acceleration is approximately 9.8 m/s2 directed downward.
2) Formulas are provided for gravitational acceleration based on Newton's law of universal gravitation, as well as kinematic equations of motion involving displacement, velocity, acceleration, and time.
3) Several example problems are worked through applying the kinematic equations to situations like objects being dropped, thrown upwards, or moving upwards/downwards together to calculate values like time, velocity, displacement, and maximum height reached.
A ball is thrown vertically upwards with an initial velocity of 29.4 m/s. It reaches its maximum height of 44.1 m after 3 seconds. After 4 seconds, the ball has fallen 4.9 m from its maximum height and is located 39.2 m above the ground, having traveled a total distance of 4.9 m during its downward motion in the 1 second since reaching maximum height.
This document summarizes key concepts about two-dimensional motion and vectors:
1) It introduces scalars, which have magnitude but no direction, and vectors, which have both magnitude and direction.
2) It describes methods for adding vectors graphically by drawing them as arrows and finding the resultant, or using trigonometry.
3) It explains projectile motion as objects moving under gravity with both horizontal and vertical components of motion that can be analyzed separately using kinematic equations.
Discusses projectile motion as two dimensional motion.
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This document provides an overview of chapter six which discusses applications of the definite integral in geometry, science, and engineering. It introduces how definite integrals can be used to calculate volume, surface area, length of a plane curve, and work done by a force. It reviews key concepts like Riemann sums and finding the area between two curves. It then explains the specific applications of using integrals to find volume of solids obtained by rotating an area about an axis, surface area of revolution, and work done by a variable force. Examples are provided for each application.
1. This document contains solutions to 6 physics problems. The first problem involves water flowing between two containers when one is heated. The second solves equations of motion for circular motion attached to a string. The third calculates the maximum distance a brick can travel when thrown at an angle. The fourth analyzes the motion of an expanding sheet on an inclined plane. The fifth derives forces and damping for a charged particle near a conducting plate. The sixth fully solves the motion of a ball attached to a string swinging from a pole.
The document summarizes the calculations to determine the number of laborers required to construct the Great Pyramid of Khufu in Egypt over a 20 year period. It provides details on the dimensions and materials of the pyramid, as well as assumptions about each laborer's daily work output. Through integrating the volume formula of each pyramid slab multiplied by density and setting up proportions based on similar triangles, the total work required is calculated. This is then divided by the work one laborer could complete in 20 years to find that approximately 110,973 laborers would have been needed.
Kinematics is the study of linear motion. Key terms include displacement, velocity, and acceleration. Displacement is the distance from a starting point, velocity is speed in a direction, and acceleration is the rate of change of velocity. Average values are calculated by total distance or displacement over total time. Instantaneous values give a clearer picture of motion at a moment in time and can be derived from graphs of displacement, velocity, and acceleration over time. When acceleration is constant, five equations can be used to describe motion with constant acceleration.
1. MMPS Project 1: The Throwing Problem
Group 6: Cameron M, Curtis C, Henry S, Melissa P, Sabina A, Axel H
October 2015
1 The Problem
During Fresher’s week, and to coincide with the 2015 Rugby World Cup, one
student makes a bet with their housemates that they are able to throw a rugby
ball over the roof of their house, run through the house, and then catch the ball
as it is on its way down in the garden. Before accepting the bet, the housemates
ask your advice as to whether this is really feasible. Is it?
To answer this problem we need to first show if it is possible using an optimal
scenario taking various assumptions. Then once we’ve established whether it is
possible in an optimal scenario we can use more appropriate values taking less
assumptions and see if it is feasible in a realistic scenario.
2 The Optimal Scenario
1. The Assumptions:
Trajectory Of Throw:
For our model to be correct we must make the assumption that the flat-
mate throws in a 2D plane and there is no wind pushing the ball off its
path, this will allow us to model the problem in 2D and make it much
simpler.
Gravity:
We will assume the acceleration due to gravity on the ball to be −9.8ms−1
and denote it as g. This is a standard value and will be used in all
calculations.
Path Through House:
We will assume that the flatmate will have a direct, straight path through
the house from the front door to the back door and also that these doors
will be opened prior to the throw.
Shape Of House:
We have assumed the house to be a symmetrical 2 floor house with walls of
height wm, and additionally a sloped roof adding a maximum additional
height of ym in the center of the house, giving a maximum height of the
house, denoted as zm. Another assumption we have made is the roof and
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2. house is clear of any obstructions such as a chimney, guttering or satellite
dish.
Ability Of the Flatmate:
We have assumed our flatmate to be 18 years old and therefore will use
average values of an 18 year old for height, speed they can throw the ball
and run. We’ve assumed they will throw the ball at 2/3 the speed of a
professionals value of roughly 60mph. Then again at 2/3 the speed for
running, which for a professional is roughly 10.6ms−1
.
Air Resistance: A rugby ball is quite wide which means that it cannot be
thrown very far, but can be thrown quite accurately. The person throwing
the ball must decide on the angle and the distance they throw it depending
on the wind present. If the wind is coming strongly from behind, the ball
should be thrown at a steeper angle and more closer to the house since
as the ball goes higher up, the wind will propel the ball further. This is
because the wind does not have any obstructions at higher altitudes and
therefore it exerts more effect on the ball. However if the wind is coming
towards the person, the ball should be thrown at a smaller angle and fur-
ther back from the house. Due to the fact of there being stronger winds
towards them, the ball will go further if it is kept low and this will only
work if the person is standing further away in order to still get it over the
roof.
Furthermore, the velocity of the ball when it is thrown must be at a high
speed for it to actually travel the length of the house. When the ball is
thrown upwards the vertical velocity must be high and for it to go across
the house it must have a high horizontal velocity also. The deceleration
due to gravity will bring the vertical velocity to zero and this will bring
the ball down to the ground in the shape of a parabolic curve. This is
why it is important to have a very high initial velocity so that that the
ball doesn’t descend too quickly onto the house rather than clearing it.
The person can also consider throwing the ball with a spiral motion in
order to increase the distance it travels. Spiralling is when the ball rotates
at a high speed along it’s horizontal axis due to the manner in which it’s
thrown. This method will work better because as the ball spirals, it will
reduce the air resistance it experiences which will increase the horizon-
tal velocity as well as the distance covered. The rugby ball will become
more stable which will allow for a smooth throw and make it easier for
the flatmate to catch it at the end. In our scenario we have decided not
to include a variable of air resistance as we feel there is no specific mathe-
matical function we can integrate into to allow us to accurately gauge the
effect of air resistance on our ball and throw. We feel this is reasonable as
to tackle this problem we are considering an ideal scenario.
Width of the Ball: The width of the rugby ball is 9.55cm which when added
into the equations would give a near negligible difference of final value. To
be exact, the difference in time in the air would be 3.57 seconds compared
to 3.61 seconds which we considered to be almost incomprehensible to the
2
3. action taking place.
2. The Values:
Height of House (w): 6.1m
Height of Roof (y): 1.9m
Total height of house + roof (z): 8.0m
Height off the ground from which the ball is thrown (a): 2.5m
Distance to run within house (L): 8.7m
Speed at which flatmate can run: 7.1ms−1
Speed at which flatmate can throw the ball (V): 17.9ms−1
3. The Model
4. The Maths:
Begin by taking Distance
T ime = V elocity and by putting in our notation we
get the equation:
L + 2x
T
= V cosθ
Rearranging this formula can give us an expression for T:
T =
L + 2x
V cosθ
3
4. Before we can go any further we need to find some equations that will give
us expressions for T. We can do this by deriving some kinematic equations
which are referred to in shorthand as S.U.V.A.T. Firstly:
v = u + at
Acceleration is defined as change of velocity over time. a = v−u
t which we
can rearrange easily to give us v = u + at.
s = ut +
at2
2
First we must derive s = u+v
2 t. This rearranges to give v = s
t from which
we get the expression s = vt. The average velocity is v+u
2 . Substituting
this in for v gives s = v+u
2 t.
Using the two previously derived equations will give us another expression:
s = 2u+at
2 t which we then simplify to give s = ut + at2
2 . Now these
expressions are ready to be used in our working.
We can solve the earlier expression using the S.U.V.A.T equation v = u+at
to get another expression for T. The lowercase t in this case denoting the
time taken for the ball to make the horizontal journey.
−V sinθ = V sinθ − 9.8T
Now this equation can be easily rearranged to find another expression for
T, as shown here:
T =
2V sinθ
9.8
As these are both expressions for T we can equate them to give the ex-
pression:
L + 2x
V cosθ
=
2V sinθ
9.8
This can then be rearranged to get an expression for x by multiplying
through by V cosθ and doing simple rearrangements, to finally give us an
expression we can name
(∗) : x =
5
49
V 2
sinθcosθ −
L
2
Since we know that Distance
T ime = V elocity gives us another equation when
we substitute in the algebraic equivalents, such that:
4
5. x
t
= V cosθ ⇒ t =
x
V cosθ
Now we’ll use another kinematic formula to get a quadratic equation which
we can then use to find a suitable value for the optimal angle to throw,
by simply inputting values for a distance to run and speed at which the
ball is thrown. The kinematic equation we will use is S = ut + at2
2 where
S denotes the height the bal has to travel before clearing the house’s
guttering, being measured as S = w − a. Then first putting the initial
values in we come to get the equation:
S = (V sinθ)(
x
V cosθ
) −
49
10
(
x
V cosθ
)2
Now we can substitute the expression for x we have from (∗) to get the
equation:
S =
5
49 V 3
sin2
θcosθ − LV sinθ
2
V cosθ
−
49
10
(
25
2401 V 4
sin2
θcos2
θ − 5
49 V 2
Lsinθcosθ + L2
4
V 2cos2θ
)
As you can see that equation is very lengthy. Hence, with quite a sub-
stantial amount of expanding out and then simplifying, we eventually get
the equation:
(
5V 2
98
)n2
+ (−S −
5V 2
98
)n + (S −
49L2
40V 2
) = 0
Now we can input our values for L and V we get a value that we have
denoted as n where n = sin2
θ, from which we can obtain the optimal
angle for which the ball should be thrown in order to both maximise the
length of time the ball is in the air and the best arch where it will fall
closest to the house thus minimising the distance the flatmate has to run.
We will also use this angle to work out the time the ball will be in the
air and the distance the flatmate needs to stand back from the house to
achieve this perfect arch.
So when we put in our values we get:
1602.05
98
n2
+ (−3.6 −
1602.05
98
)n + (3.6 −
3708.81
12816.4
) = 0
We can then take this quadratic and use the quadratic formula to solve
for a value of n
−b ±
√
b2 − 4ac
2a
⇒
12.747 ± 162.497 − 4(16.347)(3.311)
32.695
5
6. The quadratic equation above gives solution of n = −0.447. This then
works out to give the value for θ = 81.2°which we can then put back into
our earlier equations to find T and x.
x = 32.695(sin(81.2))(cos(81.2)) − 4.35 ⇒ x = 0.593m
T = 3.653sin(81.2) ⇒ T = 3.61seconds
Now if we compare this time against the time it would take the flatmate
to run the full distance L + 2x ⇔ 8.7 + 2(0.593) ⇔ 9.886m we can see if
it is feasible. The speed at which our flatmate runs is 7.1ms−1
and the
distance to cover is 9.886m, so they will cover that distance in 9.886
7.1 = 1.392
seconds, thus having plenty of time to catch the ball.
3 Conclusion
The problem poses the question of is the act physically feasible?
In the case of our optimal scenario, the answer to this is clearly Yes since the
flatmate has almost 2 full seconds after they have covered the required distance
on the ground before the ball reaches the height of their outstretched arm.
Furthermore, of course the flatmate can still catch the ball at any lower height
before it reaches the ground.
While our model may seem simplistic, and it may have been ideal to construct a
further scenario with increased variables and a more accurate model, ours fully
answers the question posed with enough mathematical evidence to state it as
correct and justified.
4 Bibliography
Effect of air resistance on rugby ball: http://envisionrugby.weebly.com/
Speed of ball throw: http://ftw.usatoday.com/2014/03/how-fast-football-
throw-nfl-combine-logan-Thomas
http://www.cbssports.com/nfl/eye-on-football/24466665/colin-kaepernick-no-longer-
has-nfl-combine-record-for-fastest-pass
Average Height of 18 year old: http://www.bbc.co.uk/news/uk-11534042
Speed of running: http://voices.nationalgeographic.com/2013/08/09/how-
fast-can-a-human-run/
Values for House Averages: http://reneweconomy.com.au/2013/how-big-is-
a-house-average-house-size-by-country-78685
We took the value for the UK on that site and square rooted them in order to
get a width of the house.
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