Angular Quantites

http://hyperphysics.phy-astr.gsu.edu/hbase/circ.html

In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined:

s = r θ

where s is the arc length.

Rotational Motion (Chapter 8)

Objectives

1. Convert angular quantities from revolutions or degrees to radians and vice versa.

2. Write the Greek symbols used to represent angular displacement, angular velocity, and angular acceleration.

3. State the meaning of the symbols used in the kinematics equations for uniformly accelerated angular motion.

4. Write from memory the equations used to describe uniformly accelerated angular motion.

5. Complete a data table using information both given and implied in word problems. Use the completed data table to solve word problems related to angular kinematics.

6. Distinguish between inertia and moment of inertia.

7. Distinguish between linear momentum and angular momentum. State and apply the law of conservation of angular momentum to solve word problems.

8. Calculate the lever arm distance and determine the magnitude and direction of the torque vector if the magnitude and direction of the net force are given.

9. Draw a free body diagram for each object in a system. Locate the forces acting on each object. Use F = ma and τ = Iα to solve for the linear or angular acceleration of each object.

10. Apply the law of conservation of angular momentum to a system where no net external torque acts. Determine the change in angular velocity of a system where the moment of inertia of the objects that make up the system changes.

11. Distinguish between translational kinetic energy and rotational kinetic energy. Apply the Law of Conservation of Energy to solve problems that involve rotational as well as translational kinetic energy.

Practice Questions

1. An Olympic diver dives off the high-diving platform. The magnitude of his momentum will be a maximum at Point

(A)A
(B)B
(C)C
(D) D

2. Which of the following can best be described as an elastic collision?
(A) A falling tree hitting the earth.
(B) A linebacker stopping a fullback.
(C) A golf club striking a golf ball.
(D) A pie in the face.

3. The Center of Mass of which of the following objects would not lie within the body itself?
(A) baseball
(B) brick
(C) frisbee
(D) paperback book

4. What is the magnitude of the impulse as shown in the following figure?

(A)100 Ns
(B)900 Ns
(C)1000 Ns
(D)1100 Ns

5. If the impulse is applied to a particle at rest, what will be its final momentum?

(A)100 kg m/s
(B)900 kg m/s
(C)1000 kg m/s
(D)1100 kg m/s

6. What is the impulse shown in the following figure?

(A)225 Ns
(B)450 Ns
(C)900 Ns
(D) 1000 Ns

7. If the impulse is applied to a 200 gm model rocket at rest, what will be its final speed?

(A) 2.25 m/s
(B) 22.5 m/s
(C) 2250 m/s
(D) 4500 m/s

8. What constant force acting over the same time as the force in Question 6 would produce the same impulse?

(A)45 N
(B) 50 N
(C) 90 N
(D)100 N

9. If each mass in the figure is 1 kg, what is the x location of the center of mass?

(A) .5
(B) .67
(C)1.0
(D) 1.33

10. If each mass in the figure is 1 kg, what is the y location of the center of mass?

(A) .33
(B) .5
(C) .67
(D) .85

11. If the mass of m1 and m3 in the figure is 1 kg and m2 is 2 kg, what is the x location of the center of mass?

(A) 1.33
(B) 1.0
(C) .67
(D) .33

12. If the mass of m1 and m3 in the figure is 1 kg and m2 is 2 kg, what is the y location of the center of mass?

(A) .75
(B) .5
(C) .33
(D) .25

13. If the mass of m1 is 1 kg and m3 is 2 kg in the figure and m2 is 3 kg, what is the x location of the center of mass?

(A) .67
(B) 1.17
(C) 1.33
(D) 1.67

14. If the mass of m1 is 1 kg and m3 is 2 kg in the figure and m2 is 3 kg, what is the y location of the center of mass?

(A).25
(B).5
(C) .67
(D) .75

15. If all the masses in the figure have a mass of 2 kg, and m2 has speed 2 m/s and m3 has a speed of 4 m/s, what is the x-component of the momentum of the center of mass?

(A)2 kg m/s
(B)4 kg m/s
(C)6 kg m/s
(D)10 kg m/s

16. If all the masses in the figure have a mass of 2 kg, and m2 has speed 2 m/s and m3 has a speed of 4 m/s, what is the y-component of the momentum of the center of mass?

(A)2 kg m/s
(B) 4 kg m/s
(C) 8 kg m/s
(D) 10 kg m/s

17. What is the magnitude of the momentum of the center of mass of the system of particles in Question 15?

(A) 4kg m/s
(B) 8 kg m/s
(C) 8.9 kg m/s
(D)10 kg m/s

18. What is the direction of the momentum of the center of mass of the particles in Question 15, as measured in degrees counter clockwise from East?

(A) 15
(B) 34
(C) 45
(D) 63

19. A 12000 kg railroad car travelling at 10 m/s strikes and couples with a 6000 kg caboose at rest. What is the speed of the final combination?

(A) 3.3 m/s
(B)5.0 m/s
(C)6.7 m/s
(D)10 m/s

20. A 12000 kg railroad car travelling at 10 m/s strikes and couples with a 6000 kg caboose at rest. What is the loss of mechanical energy in the collision?

(A) 50000 J

(B) 100000 J

(C) 200000 J

(D) 250000 J

21. A 12000 kg railroad car travelling at 10 m/s strikes and couples with a 6000 kg caboose at rest. What would be the speed of the combination of railroad cars if the caboose was initially rolling towards the other car with a speed of 2 m/s?

(A) 0 m/s

(B) 6.0 m/s

(C) 7.3 m/s

(C) 12 m/s

22. A billiard ball travelling to the right at 6m/s overtakes and strikes an identical ball travelling at 3 m/s. What is the speed of the first ball after the collision?

(A) -6 m/s

(B) -3 m/s

(C) 3 m/s

(D) 6 m/s

23. A billiard ball travelling to the right at 6m/s overtakes and strikes an identical ball travelling at 3 m/s. What is the speed of the second ball after the collision?

(A)-6 m/s

(B) -3 m/s

(C) 3 m/s

(D) 6 m/s

24. A billiard ball travelling to the right at 6m/s overtakes and strikes an identical ball travelling at 3 m/s. What is the speed of the first ball after collision if the second ball was travelling at it (instead of away from it) with a speed of 3 m/s before collision?

(A) -6 m/s

(B) -3 m/s

(C) 3 m/s

(D) 6 m/s

25. What is the speed of the second ball in Question 24 after collision?

(A) -6 m/s

(B) -3 m/s

(C) 3 m/s

(D) 6 m/s

Centre of Mass

Inelastic Collison

Kinetic Energy lost

Elastic Collision II

Elastic Collision

Conservation of Momentum

from www.physicsclassroom.com

Momentum is conserved.

force and momentum

The rate of change of momentum of an object is equal to the net force applied to it.

In order to change the momentum of an object, a force must be applied .
The time rate of change of momentum of an object is equal to the net force acting on it.

Impulse and Momentum

If momentum changes, it’s because mass or velocity change.

Most often mass doesn’t change so velocity changes and that is acceleration.

And mass x acceleration = force

Applying a force over a time interval to an object changes the momentum

Force x time interval = Impulse

Impulse = F x t or Ft = mv

What’s Momentum ?

Momentum = mass x velocity

momentum = mv

if direction is not an important factor : . . momentum = mass x speed

So, A really slow moving truck and an extremely fast roller skate can have the same momentum.

Unit: kgm/s

Linear Momentum (Chapter 7)

Objectives

1. Define linear momentum and write the mathematical formula for linear momentum from memory.

2. Distinguish between the unit of force and momentum.

3. Write Newton's Second Law of Motion in terms of momentum.

4. Define impulse and write the equation that connects impulse and momentum.

5. State the Law of Conservation of Momentum and write, in vector form, the law for a system involving two or more point masses.

6. Distinguish between a perfectly elastic collision and a completely inelastic collision.

7. Apply the laws of conservation of momentum and energy to problems involving collisions between two point masses.

8. Define center of mass and center of gravity and distinguish between the two concepts.

Practice Questions VI

1. Two men, Joel and Jerry, push against a wall. Jerry stops after 10 min, while Joel is able to push for 5 min longer. Compare the work against the wall they each do.
(A) Joel does 50% more work than Jerry.
(B) Jerry does 50% more work than Joel.
(C) Joel does 75% more work than Jerry.
(D) Neither of them do any work.

2. A simple pendulum, consisting of a mass m and a string, swings upward, making an angle θ with the vertical. The work done by the tension force is

(A) zero.

(B) mg.

(C) mg cos theta

(C) mg sin theta

3. A simple pendulum, consisting of a mass m, is attached to the end of a 1.5 m length of string. If the mass is held out horizontally, and then released from rest, its speed at the bottom is

(A) 4.4 m/s

(B) 5.4 m/s

(C) 9.8 m/s

(D) 17 m/s

4. A 4-kg mass moving with speed 2 m/s, and a 2-kg mass moving with a speed of 4 m/s, are gliding over a horizontal frictionless surface. Both objects encounter the same horizontal force, which directly opposes their motion, and are brought to rest by it. Which statement best describes their respective stopping distances?

(A) The 4-kg mass travels twice as far as the 2-kg mass before stopping.

(B) The 2-kg mass travels twice as far as the 4-kg mass before stopping.

(C) Both masses travel the same distance before stopping.

(D) The 2 kg mass travels greater than twice as far.

5. A 4-kg mass moving with speed 2 m/s and, an otherwise identical, 2-kg mass moving with a speed of 4 m/s, are gliding over a horizontal surface with friction and are brought to rest by it. Which statement best describes their respective stopping distances?

(A) The 4-kg mass travels twice as far as the 2-kg mass before stopping.

(B) The 2-kg mass travels twice as far as the 4-kg mass before stopping.

(C) Both masses travel the same distance before stopping.

(D) The 2-kg mass travels greater than twice as far.

6. A force that Object A exerts on Object B is observed over a 10-second interval, as shown on the graph. How much work did Object A do during that 10 s?

(A) Zero

(B) 12.5 J

(C) 25 J

(D) 50 J

7. A force that Object A exerts on Object B is observed over a 10-second interval, as shown on the graph. What is the average power output of A into B?

(A) O W

(B) 1.3 W

(C) 2.5 W

(D) 5 W

8. The resultant force you exert on a shopping cart, for a 10 s period, is plotted on the graph shown. How much work did you do during this 10 s interval?

(A) Zero

(B) 12.5 J

(C) 25 J

(D) -25 J

9. The resultant force you exert on a shopping cart, for a 10 s period, is plotted on the graph, shown. Which of the following statements are true?

(A) The average power input into B is greater than zero.

(B) The average power input into A is the same in the first half as the power input in the second half.

(C) The average power equals the instantaneous power.

(D) The average power is zero.

10. How much work was required to bring the 1000-kg roller coaster from Point P to rest at Point Q at the top of the 50 m peak?

(A) 32,000 J

(B) 50,000 J

(C) 245,000 J

(D) 490,000 J

11. If the roller coaster leaves Point Q from rest, how fast is it traveling at Point R?

(A) 22.1 m/s

(B) 31.3 m/s

(C) 490 m/s

(D) 980 m/s

12. What was the total work done on you by all the forces in the universe between the time just before you awoke this morning and right now?

(A) Don't have a clue

(B) Greater than zero.

(C) Zero.

(D) Can not be calculated.

13. Two cars, starting from rest at the same place, travel by different routes to the same destination. One of the cars passes the other as they drive through it. Which of the following statements will be true?

(A) The work done by friction during the trip was the same for both

(B) The total work done on both is the same.

(C) The work done by gravity is the same on both.

(D) The work done by gravity on both is positive.

14. The work done by friction, f,

(A) equals -fd, where d is the total distance moved.

(B) equals fd, where d is the total distance.

(C) can't easily be calculated because it depends on the angle between f and d.

(D) can't easily be calculated.

15. A 102 kg man climbs a 5.0 meter high stair case at constant speed. How much work does he do?

(A) 510 J

(B) 49 J

(C) 5000 J

(D) 2500 J

15. A ball is released, from rest, at the left side of the loop-the-loop, at the height shown (h = 2R). If the radius of the loop is R and there is no friction, what vertical height does the ball rise to on the other side?

(A) Less than R

(B) R

(C) 2R

(D) Greater than R