The document discusses vector concepts like displacement, velocity, and force and how to calculate their components in x and y directions. It explains how to use vectors to model projectile motion and calculate quantities like range. It also covers how forces balance in two dimensions, how to find acceleration on an inclined plane, and how to resolve forces into perpendicular components.
Week 3 OverviewLast week, we covered multiple forces acting on.docxmelbruce90096
Week 3 Overview
Last week, we covered multiple forces acting on an object. This week we will cover motion in two dimensions, inclined planes, circular motion, and rotation.
Forces in Two Dimensions (1 of 2)
So far you have dealt with single forces acting on a body or more than two forces that act parallel to each other. But in real life situations more than one force may act on a body. How are Newton's laws applied to such cases? We will restrict the forces to two dimensions.
Since force and acceleration are vectors, Newton's law can be applied independently to the X and Y-axes of a coordinate system. For a given problem you can choose a suitable coordinate system. But once a coordinate system is chosen, we have to stick with it for that problem. The example that follows shows how to find the acceleration of a body when two forces act on it at right angles to each other.
Forces in Two Dimensions (2 of 2)
To find the resultant acceleration we draw an arrow OA of length 3 units along the X-axis and then an arrow AB of length 4 units along the Y-axis. The resultant acceleration is the arrow OB with the length of 5 units. Therefore, the acceleration is 5 m/s2 in the direction of OB. Also when you measure the angle AOB with a protractor, we find it to be 53°.
The acceleration caused by the two forces is 5 m/s2 at an angle of 53°.
Uniform Circular Motion
When an object travels in a circular path at a constant speed, its motion is referred to as uniform circular motion, and the object is accelerated towards the center of the circle. If the radius of the circular path is r, the magnitude of this acceleration is ac = v2 / r, where v is its speed and ac is called the centripetal acceleration. A centripetal force is responsible for the centripetal acceleration, which constantly pulls the object towards the center of the circular path. There cannot be any circular motion without a centripetal force.
Banking
When there is a sharp turn in the road or when a turn has to be taken at a high speed as in a racetrack, the outer part of the road or the track is raised from the inner part of the track. This is called banking. It provides additional centripetal force to a turning vehicle so that it doesn't skid.
The angle of banking is kept just right so that it provides all the centripetal force required and a motorist does not have to depend on the friction force at all.
Inclined Planes
Forces on an Inclined Plane
The inclined plane is a device that reduces the force needed to lift objects. Consider the forces acting on a block on an inclined surface. The inclined surface exerts a normal force FN on the block that is perpendicular to the incline. The force of gravity, FG, points downward. If there is no friction, the net force, Fnet, acting on the block is the resultant of FN and FG. By Newton's second law the net force must point down the incline because the block moves only along the incline and not perpendicular to it.
The vector triangle shows .
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page.docxwillcoxjanay
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 1 -
The Two-Dimensional Motion of a Projectile
Equipment: air table strobe (furnished by instructor)
meter sticks
Each group will be given a strobe picture of a two-dimensional motion that simulates
the motion of a projectile. By “projectile” it is meant any object thrown or dropped,
which accelerates under the influence of the force of gravity. If the object is given an
initial horizontal component of velocity it will execute a two-dimensional motion. That
is, its path through space will require a specification of two numbers at each point
(vertical and horizontal coordinate) for a complete description. The motions we have
been studying so far , in contrast, have been one-dimensional in that, to describe the
trajectory only one position coordinate need be specified at each point. All
motions take place in three dimensional space, of course, but two and one-dimensional
motions are confined to planes and lines respectively.
The motion depicted is only a simulation of a true projectile as it represents the motion
of a puck sliding over the surface of a tilted air table. The puck is projected up the incline
at an angle so that its velocity has both horizontal and vertical velocity components. A
true projectile is in vertical free fall. The motion studied here will differ from a true
projectile in one respect: the vertical acceleration will be less than g . However, the
essential features of the air table motion are representative of what one would observe for
a true projectile.
Velocity Components
An object executing a two dimensional motion as described above, will have two
position variables that change with time. That is, at each point along its trajectory the
object will have (in general) different vertical and horizontal coordinates. At every instant
we can then define the rate at which the vertical (or y) coordinate is changing and call
this the vertical (or y) velocity, and we can define a corresponding quantity for the
horizontal (or x) coordinate. We call this second quantity the horizontal (or x) velocity.
These quantities are also referred to as components of the overall velocity vector.
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 2 -
One way to picture a motion in two dimensions is to think of it as two, one-dimensional
motions. In the case of a projectile, that would mean, one vertical and the other
horizontal. Each coordinate, vertical and horizontal, traces out a trajectory (position
versus time). The horizontal trajectory can be thought of as the shadow of the object, as it
moves, projected onto the ground. The vertical trajectory is a shadow projected onto a
wall. The respective velocity components represent the velocities of the respective
shadows. We can think along similar lines as we seek to describe the motion of the air
table puck replacing vertical with the “up the inc ...
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Week 3 OverviewLast week, we covered multiple forces acting on.docxmelbruce90096
Week 3 Overview
Last week, we covered multiple forces acting on an object. This week we will cover motion in two dimensions, inclined planes, circular motion, and rotation.
Forces in Two Dimensions (1 of 2)
So far you have dealt with single forces acting on a body or more than two forces that act parallel to each other. But in real life situations more than one force may act on a body. How are Newton's laws applied to such cases? We will restrict the forces to two dimensions.
Since force and acceleration are vectors, Newton's law can be applied independently to the X and Y-axes of a coordinate system. For a given problem you can choose a suitable coordinate system. But once a coordinate system is chosen, we have to stick with it for that problem. The example that follows shows how to find the acceleration of a body when two forces act on it at right angles to each other.
Forces in Two Dimensions (2 of 2)
To find the resultant acceleration we draw an arrow OA of length 3 units along the X-axis and then an arrow AB of length 4 units along the Y-axis. The resultant acceleration is the arrow OB with the length of 5 units. Therefore, the acceleration is 5 m/s2 in the direction of OB. Also when you measure the angle AOB with a protractor, we find it to be 53°.
The acceleration caused by the two forces is 5 m/s2 at an angle of 53°.
Uniform Circular Motion
When an object travels in a circular path at a constant speed, its motion is referred to as uniform circular motion, and the object is accelerated towards the center of the circle. If the radius of the circular path is r, the magnitude of this acceleration is ac = v2 / r, where v is its speed and ac is called the centripetal acceleration. A centripetal force is responsible for the centripetal acceleration, which constantly pulls the object towards the center of the circular path. There cannot be any circular motion without a centripetal force.
Banking
When there is a sharp turn in the road or when a turn has to be taken at a high speed as in a racetrack, the outer part of the road or the track is raised from the inner part of the track. This is called banking. It provides additional centripetal force to a turning vehicle so that it doesn't skid.
The angle of banking is kept just right so that it provides all the centripetal force required and a motorist does not have to depend on the friction force at all.
Inclined Planes
Forces on an Inclined Plane
The inclined plane is a device that reduces the force needed to lift objects. Consider the forces acting on a block on an inclined surface. The inclined surface exerts a normal force FN on the block that is perpendicular to the incline. The force of gravity, FG, points downward. If there is no friction, the net force, Fnet, acting on the block is the resultant of FN and FG. By Newton's second law the net force must point down the incline because the block moves only along the incline and not perpendicular to it.
The vector triangle shows .
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page.docxwillcoxjanay
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 1 -
The Two-Dimensional Motion of a Projectile
Equipment: air table strobe (furnished by instructor)
meter sticks
Each group will be given a strobe picture of a two-dimensional motion that simulates
the motion of a projectile. By “projectile” it is meant any object thrown or dropped,
which accelerates under the influence of the force of gravity. If the object is given an
initial horizontal component of velocity it will execute a two-dimensional motion. That
is, its path through space will require a specification of two numbers at each point
(vertical and horizontal coordinate) for a complete description. The motions we have
been studying so far , in contrast, have been one-dimensional in that, to describe the
trajectory only one position coordinate need be specified at each point. All
motions take place in three dimensional space, of course, but two and one-dimensional
motions are confined to planes and lines respectively.
The motion depicted is only a simulation of a true projectile as it represents the motion
of a puck sliding over the surface of a tilted air table. The puck is projected up the incline
at an angle so that its velocity has both horizontal and vertical velocity components. A
true projectile is in vertical free fall. The motion studied here will differ from a true
projectile in one respect: the vertical acceleration will be less than g . However, the
essential features of the air table motion are representative of what one would observe for
a true projectile.
Velocity Components
An object executing a two dimensional motion as described above, will have two
position variables that change with time. That is, at each point along its trajectory the
object will have (in general) different vertical and horizontal coordinates. At every instant
we can then define the rate at which the vertical (or y) coordinate is changing and call
this the vertical (or y) velocity, and we can define a corresponding quantity for the
horizontal (or x) coordinate. We call this second quantity the horizontal (or x) velocity.
These quantities are also referred to as components of the overall velocity vector.
Two Dimensional Motion…, copyright Doug Bradley-Hutchison page - 2 -
One way to picture a motion in two dimensions is to think of it as two, one-dimensional
motions. In the case of a projectile, that would mean, one vertical and the other
horizontal. Each coordinate, vertical and horizontal, traces out a trajectory (position
versus time). The horizontal trajectory can be thought of as the shadow of the object, as it
moves, projected onto the ground. The vertical trajectory is a shadow projected onto a
wall. The respective velocity components represent the velocities of the respective
shadows. We can think along similar lines as we seek to describe the motion of the air
table puck replacing vertical with the “up the inc ...
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
1. Vector Objectives
1. Add and subtract displacement vectors to describe
changes in position.
2. Calculate the x and y components of a displacement,
velocity, and force vector.
3. Write a velocity vector in polar and x-y coordinates.
4. Calculate the range of a projectile given the initial velocity
vector.
5. Use force vectors to solve two-dimensional equilibrium
problems with up to three forces.
6. Calculate the acceleration on an inclined plane when
given the angle of incline.
3. 7.1 Vectors and Direction
Key Question:
How do we accurately
communicate length
and distance?
*Students read Section 7.1 AFTER Investigation 7.1
4. Vectors and Direction
A scalar is a quantity that
can be completely
described by one value: the
magnitude.
You can think of magnitude
as size or amount,
including units.
5. Vectors and Direction
A vector is a quantity that
includes both magnitude
and direction.
Vectors require more than
one number.
The information “1
kilometer, 40 degrees east
of north” is an example of a
vector.
6. Vectors and Direction
In drawing a vector as an
arrow you must choose a
scale.
If you walk five meters
east, your displacement
can be represented by a 5
cm arrow pointing to the
east.
7. Vectors and Direction
Suppose you walk 5 meters east,
turn, go 8 meters north, then
turn and go 3 meters west.
Your position is now 8 meters
north and 2 meters east of
where you started.
The diagonal vector that
connects the starting position
with the final position is called
the resultant.
8. Vectors and Direction
The resultant is the sum of two
or more vectors added together.
You could have walked a shorter
distance by going 2 m east and
8 m north, and still ended up in
the same place.
The resultant shows the most
direct line between the starting
position and the final position.
9.
10.
11. Calculate a resultant vector
An ant walks 2 meters West, 3 meters North,
and 6 meters East.
What is the displacement of the ant?
12. Finding Vector Components
Graphically
Draw a
displacement
vector as an arrow
of appropriate
length at the
specified angle.
Mark the angle and
use a ruler to draw
the arrow.
13.
14. Finding the Magnitude of a Vector
When you know the x- and y- components of a vector, and
the vectors form a right triangle, you can find the
magnitude using the Pythagorean theorem.
17. Calculate vector magnitude
A mail-delivery robot
needs to get from where it
is to the mail bin on the
map.
Find a sequence of two
displacement vectors that
will allow the robot to
avoid hitting the desk in
the middle.
18. Projectile Motion and the Velocity
Vector
Any object that is
moving through the air
affected only by gravity
is called a projectile.
The path a projectile
follows is called its
trajectory.
19. Projectile Motion and the Velocity
Vector
The trajectory of a
thrown basketball
follows a special type of
arch-shaped curve called
a parabola.
The distance a projectile
travels horizontally is
called its range.
20.
21. Projectile Motion and the Velocity
Vector
The velocity vector (v) is a way
to precisely describe the speed
and direction of motion.
There are two ways to
represent velocity.
Both tell how fast and in what
direction the ball travels.
22. Calculate magnitude
Draw the velocity vector v
= (5, 5) m/sec and
calculate the magnitude
of the velocity (the
speed), using the
Pythagorean theorem.
23. Components of the Velocity Vector
Suppose a car is driving 20
meters per second.
The direction of the vector
is 127 degrees.
The polar representation
of the velocity is v = (20
m/sec, 127°).
24. Calculate velocity
A soccer ball is kicked at a speed of 10 m/s and an angle
of 30 degrees.
Find the horizontal and vertical components of the ball’s
initial velocity.
25. Adding Velocity Components
Sometimes the total velocity of an object is a combination of
velocities.
One example is the motion of a boat on a river.
The boat moves with a certain velocity relative to the
water.
The water is also moving with another velocity relative to
the land.
27. Calculate velocity components
An airplane is moving at a velocity of 100 m/s in a direction 30
degrees NE relative to the air.
The wind is blowing 40 m/s in a direction 45 degrees SE relative
to the ground.
Find the resultant velocity of the airplane relative to the
ground.
28. Projectile Motion
When we drop a ball
from a height we know
that its speed increases
as it falls.
The increase in speed is
due to the acceleration
gravity, g = 9.8 m/sec2.
Vx
Vy
x
y
29. Horizontal Speed
The ball’s horizontal velocity
remains constant while it
falls because gravity does not
exert any horizontal force.
Since there is no force, the
horizontal acceleration is
zero (ax = 0).
The ball will keep moving to
the right at 5 m/sec.
30. Horizontal Speed
The horizontal distance a projectile moves can be
calculated according to the formula:
31. Vertical Speed
The vertical speed (vy) of the
ball will increase by 9.8 m/sec
after each second.
After one second has passed,
vy of the ball will be 9.8 m/sec.
After the 2nd second has
passed, vy will be 19.6 m/sec
and so on.
32.
33. Calculate using projectile motion
A stunt driver steers a car off
a cliff at a speed of 20
meters per second.
He lands in the lake below
two seconds later.
Find the height of the cliff
and the horizontal distance
the car travels.
34. Projectiles Launched at an Angle
A soccer ball kicked
off the ground is
also a projectile, but
it starts with an
initial velocity that
has both vertical and
horizontal
components.
*The launch angle determines how the initial velocity
divides between vertical (y) and horizontal (x) directions.
35. Steep Angle
A ball launched at
a steep angle will
have a large
vertical velocity
component and a
small horizontal
velocity.
36. Shallow Angle
A ball launched at
a low angle will
have a large
horizontal velocity
component and a
small vertical one.
37. Projectiles Launched at an Angle
The initial velocity components of an object launched at a velocity vo
and angle θ are found by breaking the velocity into x and y
components.
38. Range of a Projectile
The range, or horizontal distance, traveled by a
projectile depends on the launch speed and the launch
angle.
39. Range of a Projectile
The range of a projectile is calculated from the
horizontal velocity and the time of flight.
40. Range of a Projectile
A projectile travels farthest when launched at 45
degrees.
41. Range of a Projectile
The vertical velocity is responsible for giving the
projectile its "hang" time.
42. "Hang Time"
You can easily calculate your own hang time.
Run toward a doorway and jump as high as you can, touching the wall or door frame.
Have someone watch to see exactly how high you reach.
Measure this distance with a meter stick.
The vertical distance formula can be rearranged to solve for time:
43. Projectile Motion and the Velocity
Vector
Key Question:
Can you predict the landing spot of a projectile?
*Students read Section 7.2 BEFORE Investigation 7.2
45. In order to solve “x” we must know
“t”
Y = vot – ½ g t2
2y = g t2
vot = 0 (zero)
Y = ½ g t2
t2 = 2y
g
t = 2y
g
46. Forces in Two Dimensions
Force is also represented in x-y components.
47. Force Vectors
If an object is in
equilibrium, all of the
forces acting on it are
balanced and the net force
is zero.
If the forces act in two
dimensions, then all of the
forces in the x-direction
and y-direction balance
separately.
48. Equilibrium and Forces
It is much more difficult
for a gymnast to hold his
arms out at a 45-degree
angle.
To see why, consider that
each arm must still
support 350 newtons
vertically to balance the
force of gravity.
49. Forces in Two Dimensions
Use the y-component to find the total force in the
gymnast’s left arm.
50. Forces in Two Dimensions
The force in the right arm must also be 495 newtons
because it also has a vertical component of 350 N.
51. Forces in Two Dimensions
When the gymnast’s arms
are at an angle, only part of
the force from each arm is
vertical.
The total force must be
larger because the vertical
component of force in each
arm must still equal half his
weight.
52. Forces and Inclined Planes
An inclined plane is a straight surface, usually with
a slope.
Consider a block sliding
down a ramp.
There are three forces that
act on the block:
gravity (weight).
friction
the reaction force
acting on the block.
53. Forces and Inclined Planes
When discussing forces, the word “normal” means
“perpendicular to.”
The normal force acting
on the block is the
reaction force from the
weight of the block
pressing against the
ramp.
54. Forces and Inclined Planes
The normal force on
the block is equal
and opposite to the
component of the
block’s weight
perpendicular to the
ramp (Fy).
55. Forces and Inclined Planes
The force parallel to
the surface (Fx) is
given by
Fx = mg sinθ.
56.
57. Acceleration on a Ramp
Newton’s second law can be used to calculate the
acceleration once you know the components of all the
forces on an incline.
According to the second law:
a = F
m
Force (kg . m/sec2)
Mass (kg)
Acceleration
(m/sec2)
58. Acceleration on a Ramp
Since the block can only accelerate along the ramp, the force that
matters is the net force in the x direction, parallel to the ramp.
If we ignore friction, and substitute Newtons' 2nd Law, the net
force is:
Fx =
a =
m sin θ
g
F
m
59. Acceleration on a Ramp
To account for friction, the horizontal component of
acceleration is reduced by combining equations:
Fx = mg sin θ - m mg cos θ
60. Acceleration on a Ramp
For a smooth surface, the coefficient of friction (μ) is
usually in the range 0.1 - 0.3.
The resulting equation for acceleration is:
61. Calculate acceleration on a ramp
A skier with a mass of 50 kg is on a hill making an angle of
20 degrees.
The friction force is 30 N.
What is the skier’s acceleration?