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# 6 - Forces & eqm

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### 6 - Forces & eqm

1. 1. FORCES & EQUILIBRIUM 1. Revise the effects of balanced and unbalanced forces on the motion of an object. 2. Recognise that an object accelerates in response to the resultant force and that F = ma (Newton’s second law) 3. Recognise the resultant force on an object as being the sum of the individual forces acting or the sum of the components 4. Know the definition of torque and use it to calculate torque in a variety of everyday situations. 5. Explain that for a body to be in rotational equilibrium, the sum of the torques acting around any point equals zero and solve problems using this principle. 6. Explain that for a body to be in translational equilibrium, the sum of the forces acting in any direction equals zero and solve problems using this principle. 7. Explain that for a body to be in equilibrium then the conditions for both rotational and translational equilibrium coexist and be able to solve problems based on everyday situations, using these principles. Embedded SLO’s 1. Explain the difference between mass and weight 2. Recognise that forces are vectors and as such they can be resolved into components Read p.105 to 117
2. 2. Revision FORCE AND ACCELERATION First Law An object is reluctant to change its state of motion. An object will remain at rest or continue to travel at a steady speed unless acted upon by an unbalanced force. Second Law An object will accelerate in the direction of an unbalanced force according to the equation: F = ma Third Law For every action there is an equal and opposite reaction Examples 1. Use Newton’s laws to explain why during heavy braking (in a car) the seat belt digs into your chest.
3. 3. Revision FORCE AND ACCELERATION First Law An object is reluctant to change its state of motion. An object will remain at rest or continue to travel at a steady speed unless acted upon by an unbalanced force. Second Law An object will accelerate in the direction of an unbalanced force according to the equation: F = ma Third Law For every action there is an equal and opposite reaction Examples 1. Use Newton’s laws to explain why during heavy braking (in a car) the seat belt digs into your chest. As a passenger, you are reluctant to change your state of motion. You would continue to travel at a steady speed unless acted upon by an unbalanced force. Friction from the brakes acts in the reverse direction. This force is transferred via the seat belt to your chest (digging in). It is this force that slows you down. - Newton’s 1st & 2nd law
4. 4. 2. A sky rocket of mass 0.1 kg accelerates upwards at 10 ms-2. What is the unbalanced force that causes this acceleration. 3. A cricketer throws a ball with initial acceleration of 90 ms-2. The force exerted is 30 N. Calculate the mass of the ball. 4. A force of 60N is exerted on a car of mass 1000kg. If the frictional forces acting on the car are 10N then calculate the car’s acceleration.
5. 5. RESULTANT FORCE • Force is a vector quantity • The resultant force is the sum of all the force vectors acting on a body. • The resultant force is the single force which has the same effect as the combination of forces. Examples 1. Find the resultant force in each of the following situations 2. In each instance draw a vector diagram that shows how the resultant force has been determined: 1 2 5N 8N 12 N 10 N
6. 6. 3 Calculate the resultant force in the following situation: Calculate the force that is effectively pushing the car down the slope Ex.9A Q.1 & 2 4 (hint: this is a component of one of the forces) Support force 50N Force due to gravity 55N 25o Draw a vector triangle showing the relationship between these three forces.
7. 7. 5 A 450 g trolley rests on a rough table. A horizontal force is applied by the hanging masses and as a , the trolley travels 0.60 m from rest and reaches a final speed of 2.07 ms-1. [adapted from CR p54 Q.2 & 3] a Pulley m2 = 450 g Trolley a Hanging masses m1 = 250 g (a) Use vf2 = vi2 + 2ad to calculate the acceleration of the mass. (b) Label the horizontal force on the diagram and calculate its size. (This is the tension force and causes the acceleration of the trolley. (c) Calculate the size of the force that causes m1’s acceleration. (Hint: Use a vector diagram to show the forces acting on m1. Ex.9A Q.3 to 5
8. 8. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N 30o 1000 N
9. 9. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N The component of the 1000 N force that acts in the direction in which the cart moves is shown below: 30o 1000 N
10. 10. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N The component of the 1000 N force that acts in the direction in which the cart moves is shown below: 1000.cos30o 30o 1000 N
11. 11. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N The component of the 1000 N force that acts in the direction in which the cart moves is shown below: The trolley accelerates in response to this force 1000.cos30o 30o 1000 N
12. 12. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N The component of the 1000 N force that acts in the direction in which the cart moves is shown below: The trolley accelerates in response to this force 1000.cos30o 30o 1000 N Calculating the trolley’s acceleration:
13. 13. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N The component of the 1000 N force that acts in the direction in which the cart moves is shown below: The trolley accelerates in response to this force 1000.cos30o 30o 1000 N Calculating the trolley’s acceleration: a = F = 1000.cos30o = 43.3 ms-2 m 20
14. 14. RESULTANT AS THE SUM OF THE COMPONENT FORCES A force of 1000 N acts at an angle of 30o to the direction of a cart’s motion. What is the force that acts in the direction of the cart’s motion? Direction of cart’s motion 20 Kg 30o (axis) 1000 N Remember that a vector can be expressed as the sum of two components that are drawn at right angles to each other. The component of the 1000 N force that acts in the direction in which the cart moves is shown below: The trolley accelerates in response to this force 1000.cos30o 30o 1000 N Calculating the trolley’s acceleration: a = F = 1000.cos30o = 43.3 ms-2 m 20
15. 15. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N
16. 16. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion:
17. 17. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o
18. 18. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o
19. 19. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o 30o 1000 N
20. 20. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o 30o 1000.cos30o 1000 N
21. 21. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o 30o 1000.cos30o The trolley accelerates in response to both of 1000 N these forces
22. 22. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o 30o 1000.cos30o The trolley accelerates in response to both of 1000 N these forces Calculating the trolley’s acceleration:
23. 23. Now consider that a second force of 1000 N is introduced and that this force also acts at an angle of 30o to the direction of the cart’s motion. What is the force that acts in the direction of the cart’s motion? 1000 N 30o Direction of cart’s motion 20 Kg 30o (axis) 1000 N There are now two components which act in the direction of the cart’s motion: 1000 N 30o 1000.cos30o 30o 1000.cos30o The trolley accelerates in response to both of 1000 N these forces Calculating the trolley’s acceleration: a = F = 2 x1000.cos30o = 86.6 ms-2 m 20
24. 24. Examples 1. A large container ship is being towed out to sea by two tug boats. Each boat exerts a force of 150000 N and pulls on the ship at an angle of 20o. The ship has a mass of 1000 tonnes. (1 tonne = 1000 kg). 20o 20o Tug boats The ship is initially at rest. Calculate the acceleration of the ship.
25. 25. 2. A climber hangs off a bar with both arms. Each arm is at an angle of 45o to the vertical. The mass of the climber is 65 kg. Calculate the tension force in each arm when the climber is simply hanging stationery. Exercises: “Vectors & Components”
26. 26. WHAT IS TORQUE? Torque is a force which causes rotation (i.e. it is a turning force) Examples Because a force is applied that causes rotation, a torque is being used in each case Exercises: “Talking about torque”
27. 27. WHAT IS TORQUE? Torque is a force which causes rotation (i.e. it is a turning force) Examples Because a force is applied that causes rotation, a torque is being used in each case Torque = force applied x distance between the point of application and the pivot. = F.r⊥ Exercises: “Talking about torque”
28. 28. WHAT IS TORQUE? Torque is a force which causes rotation (i.e. it is a turning force) Examples Because a force is applied that causes rotation, a torque is being used in each case Torque = force applied x distance between the point of application and the pivot. = Torque in Newton-metres = F.r⊥ Where (Nm) F = Force applied (N) r⊥ = perpendicular distance Exercises: “Talking about torque”
29. 29. WHAT IS TORQUE? Torque is a force which causes rotation (i.e. it is a turning force) Examples Because a force is applied that causes rotation, a torque is being used in each case Torque = force applied x distance between the point of application and the pivot. = Torque in Newton-metres = F.r⊥ Where (Nm) F = Force applied (N) r⊥ = perpendicular distance r⊥ implies that the force and the distance must be perpendicular to each other. Exercises: “Talking about torque”
30. 30. Exercises TALKING ABOUT TORQUE A Torque in B is greater than torque in A because B C Torque in C is greater than torque in B because Demo: Torque wrench vs spanner
31. 31. Demo: The pattern of wear on chainrings can show where Examples in the pedal cycle maximum torque is applied. 1. The diagram below shows a bicycle crank and chain. On the crank the force and the distance between the point of application of the force and the pivot are perpendicular to each other. Calculate the torque applied in this situation: 80 kg An 80 kg cyclist stands on the pedal whilst the crank arm is in the position shown 175 mm F 2. The crank arm turns through an angle of 30o below the horizontal and the cyclist is still in a standing position. (i) Draw the situation that exists now. (ii) Calculate the torque applied in this situation. (Hint: You could find the component of the force that is perpendicular to the crank arm first)
32. 32. ROTATIONAL EQUILIBRIUM Definition A system that is not accelerating in rotational motion is said to be in rotational equilibrium. For a system to be in rotational equilibrium the following situation needs to exist: Σ any direction = 0 where = torque “The clockwise and anti-clockwise torques must be equal in size to each other” Σ clockwise =Σ anti-clockwise Bicycle crank - opposing torques Observation 10 N 10 N http://www.walter-fendt.de/ph14e/lever.htm
33. 33. Example 1. A mother and her daughter were playing on a see-say at their local park. Mum has a mass of 55 kg and the daughter is much lighter at 35 kg. The see-saw measures 2.5 m from the pivot to each end. The daughter sits at one end of the see-saw. How far from the other side of the pivot must the mother sit in order to achieve balance? 55 kg 35 kg d 2.5 m 2. A 70 kg painter stands on a 10 kg plank that is supported by two saw-horses. The plank is 4 m long and is supported 500 mm from each end. The painter stands 1 m from saw-horse A as shown. The weight of the plank acts at the centre of mass of the plank. There are 2 support forces, F1 and F2. Calculate the size F1 70 kg of F2. F2 A B Fgrav Ex.9B Q.4 to 6 Plank pbms - in finder
34. 34. TRANSLATIONAL EQUILIBRIUM Definition A system that is not accelerating (at rest or travelling at a constant speed) is said to be in translational equilibrium. For a system to be in translational equilibrium, the sum of the forces along any axis must equal zero. ΣFany direction = 0 When vectors add to zero there is no resultant ~ ~ Demo - Ping Pong ball/Hair dryer There are 3 forces acting on the ball as shown by the free - body force diagram: Lift Ping pong ball Thrust Hair dryer Gravity Adding these three vectors gives a zero resultant:
35. 35. TRANSLATIONAL EQUILIBRIUM Definition A system that is not accelerating (at rest or travelling at a constant speed) is said to be in translational equilibrium. For a system to be in translational equilibrium, the sum of the forces along any axis must equal zero. ΣFany direction = 0 When vectors add to zero there is no resultant ~ ~ Demo - Ping Pong ball/Hair dryer There are 3 forces acting on the ball as shown by the free - body force diagram: Lift Ping pong ball Thrust Hair dryer Gravity Adding these three vectors gives a zero resultant:
36. 36. TRANSLATIONAL EQUILIBRIUM Definition A system that is not accelerating (at rest or travelling at a constant speed) is said to be in translational equilibrium. For a system to be in translational equilibrium, the sum of the forces along any axis must equal zero. ΣFany direction = 0 When vectors add to zero there is no resultant ~ ~ Demo - Ping Pong ball/Hair dryer There are 3 forces acting on the ball as shown by the free - body force diagram: Lift Ping pong ball Thrust Hair dryer Gravity Adding these three vectors gives a zero resultant:
37. 37. TRANSLATIONAL EQUILIBRIUM Definition A system that is not accelerating (at rest or travelling at a constant speed) is said to be in translational equilibrium. For a system to be in translational equilibrium, the sum of the forces along any axis must equal zero. ΣFany direction = 0 When vectors add to zero there is no resultant ~ ~ Demo - Ping Pong ball/Hair dryer There are 3 forces acting on the ball as shown by the free - body force diagram: Lift Ping pong ball Thrust Hair dryer Gravity Adding these three vectors gives a zero resultant:
38. 38. Examples 1. The diagram below shows a birds eye view of a sling shot which is ready to launch a stone contained in its pouch. 26.6 N 20o 50 N 20o 26.6 Show that the stone is in translational equilibrium using a vector diagram, drawn to scale. 2. The diagram below shows a painter who is standing still on a plank that is supported at two points A and B. (a) Calculate the weight of the painter (Fgrav). 70 kg 500 N (b) Calculate the force acting at point B A B Fgrav (c) Name this force. Ex.9B Q.1 to 3
39. 39. Since we often deal with horizontal and vertical forces: Sum of Forces acting upwards = Sum of Forces acting downwards ΣFup = ΣFdown Sum of Forces acting to the left = Sum of Forces acting to the right ΣFleft = ΣFright 3. A mass is suspended by two strings that are 20o 20o fixed to the ceiling. Use a vector diagram to 70 N 70 N calculate the weight of the mass, W. Mass W
40. 40. A BODY IN EQUILIBRIUM IS IN BOTH TRANSLATIONAL AND ROTATIONAL EQUILIBRIUM Examples 1. Below is a picture of two children on a see saw which is in equilibrium. One child has a mass of 30 kg, whilst the other child has an unknown mass. The distances from the pivot (or fulcrum) are shown in the picture. 30 kg m=? 1.5 m 2.5 m (a) In terms of the children and the see saw, explain what is meant by the term “rotational equilibrium” __________________________________________________________________________ __________________________________________________________________________ (b) Calculate the mass, m of the child at the far end of the see saw. __________________________________________________________________________ __________________________________________________________________________ 2. A wheelbarrow filled with soil has a mass of 30 kg (this can be considered to act at the centre of mass). The dimensions of the barrow are shown in the diagram. Calculate the effort required to lift the barrow. ____________________________________ Centre of Mass . ____________________________________ ____________________________________ 0.7 m ____________________________________ 1.3 m Ex.9B Q.7 to 12
41. 41. 3. A Y12 physics experiment is set up as shown in the diagram. A student uses masses as described in the information below and marks angles on the paper as indicated in the diagram. The knot that connects the three strings is not moving. Show using a vector diagram drawn to scale that tension forces T1, T2 and T3 are in translational equilibrium. m1 = 0.1000 kg T1 Knot T3 m2 = 0.1200 kg m3 = 0.0833 kg 25o 20o let g = 10 ms-2 T2 m1 m3 m2 Sheet of paper
42. 42. MASS AND WEIGHT The difference between the two • Mass is a measure of the amount of matter in an object. (unit: kilogram, kg) • Weight is the force due to gravity acting on that object. (unit: Newton, N) Weight, Fw or Fgrav, is proportional to the mass, m of the object. (Fw α m) Fw = mg where g is a constant (gravitational acceleration) where g = 10 ms-2 (usually considered to be negative when direction is relevant) Note g is often referred to as gravitational acceleration since any free-falling object falls with acceleration equivalent to g (usually rounded to 10 ms-2) It is sometimes convenient to refer to g as the gravitational constant as it is also the force per unit mass on an object. (in fact g = 10 ms-2 = 10 Nkg-1) g depends on how strong the gravitational field is around the planet/moon. The gravitational field strength depends on the size of the planet/moon. Different planets/moons have different values of g. For example the Earth’s moon has a much smaller value of g than Earth. Example On Earth a 50 kg mass will have a weight force of 500 N acting on it.
43. 43. Exercises VECTORS & COMPONENTS y Any vector can be expressed as the sum of 2 vectors at right angles x to each other. The diagram shows the components of a vector in the x and y axes. Components could also be vertical or horizontal. IN FACT components could be determined in any axis or any direction The two components add to give the original vector, v. v is also could also be called the resultant of the vector addition. Calculate the components of the vectors shown along the axes/lines given (drawn as dotted lines and draw them on the diagram given. Show your calculation below the diagram. 100 m 1 60 ms -1 2 20o 75o Horizontal __________________________ __________________________ __________________________ __________________________ __________________________ __________________________
44. 44. 25 ms-1 2 km 50 ms-1 Lakewood Drive 6. The ceiling beam: A ceiling beam of mass 250 kg and length 7 m is pitched at an angle of 20o and supported at each end by timber framing. (a) Draw a diagram of the situation in the space provided: (b) Mark the point at which we can consider the force due gravity acts and calculate this force. ________________________________________ (c) Explain why this beam has a tendency to slide “downhill” and state what must be done to prevent this from happening. ______________________________________________________________ (d) Calculate the size of this “downhill” force. ______________________________________________________________
45. 45. 12 PHYSICS FORCES ASSIGNMENT Name 1. Coming into the summer sports season the ground staff have been requested to prepare the cricket wicket for the forthcoming games. The following diagram represents the efforts of the ground staff in pushing the roller. The force exerted down the handle of the roller is 200 N. 200 N 30° Use the information in the diagram to answer the question that follows. (a)Draw a labelled force diagram to show the four forces acting on the roller. (b)Show that the horizontal component of the push on the roller is 173 N.
46. 46. (c) If the roller is being pushed at a constant speed along the wicket, determine the magnitude of the force opposing the motion of the roller and explain your reasoning. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ (d) The vertical component of the push on the roller is 100 N. The mass of the roller is 100 kg. Calculate the value of the reaction force on the roller. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ (e) The length of the wicket between the stumps is 20 m. Calculate the work done in pushing the roller eight times along the pitch. ___________________________________________________________________________ ___________________________________________________________________________ __________________________________________________________________________ 2. A wheelbarrow has a mass of 15 kg and contains 50 kg of soil. The centre of mass of the load is shown on the diagram and the pivot is also illustrated. Load and effort forces are drawn and labelled on the diagram. Calculate the effort required to lift the load. ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________ ___________________________________________
47. 47. 3. A body-building machine is drawn below. Load, effort and pivot are shown. Calculate the effort required to lift the 25 kg mass. The bar has mass of 5 kg. The centre of mass of the bar and masses (combined) is shown on the diagram. Calculate the effort needed to lift the mass. 317 mm C of M 483 mm Effort 25 kg 5 kg bar Load ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4. A uniform girder is 3 m long and weighs 700 N. It is carried horizontally on the shoulders of two men who support it at points 30 cm and 60 cm from each end. Find the load carried by each man. ___________________________________________________________________ ___________________________________________________________________
48. 48. [p144 “Physics in Action”] 5. Find the centre of mass of a set of barbells, set up with 10 kg on one end and 30 kg on the other. The central bar is of negligible mass and is 4.0 m long. To solve this, we could assign a centre of mass, x metres from one end and use the fact that the barbells would balance at the centre of mass. • Bars or Dumbells balance at their centre of mass. • The weight of an object acts at is centre of mass ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________