Homework 10A (ω and α as vectors)




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Homework 10A (ω and α as vectors)


1. Shown below is a racecar driving counterclockwise around a circular track in the Cartesian plane. The radius of the racetrack is 250m, and the car is moving at 130mi/hr.

(a) What is the angular velocity vector of the car?

(b) During a 5.0s time interval, the car slows to 110mi/hr. What are the final angular velocity vector and the angular acceleration vector?




2. A pottery wheel rotates clockwise in the x-y plane, as shown below. The pottery wheel is turned on at t = 0, and at t = 2.0s, the pottery wheel is rotating at 200rpm. Assuming that the angular acceleration of the pottery wheel during this time interval is constant, find

(a) the angular acceleration vector

(b) the angular velocity vector as a function of time




y


ω



x

Homework 10B (cross-product)


3. For the two cases shown below, express AxB and BxA in unit-vector form.


z

a
B

z
. b.


A



120°

6m

3m


x

y

4. Evaluate the cross product UxV for the following cases:

a. U = 4i + 7j, V = –12j + 8k

b. U = –3j + 5k, V = –5i – 12k

c. U = 6i – 9k, V = –3i + 2j
Homework 10C (torque as cross-product)

5. The solid sphere shown below rotates clockwise about an axis through its center, and perpendicular to the page. The radius of the sphere is R = 7.5m, the mass is M = 350kg, and a 300N force is applied 3.0m from the center in the direction shown.

(a) What are the magnitude and direction of the torque on the cylinder?

(b) What is the vector expression for the torque on the cylinder?

(c) What is the vector expression for the resulting angular acceleration?


y



F = 300N


30°


3m


x

6. In the diagram below, a 20N force acts on a hollow sphere with mass M = 5.0kg and radius R = 30cm. The sphere rotates about an axis through its center, and perpendicular to the page. Assuming that the sphere starts from rest, find

(a) the torque vector

(b) the angular acceleration vector

(c) the angular velocity vector as a function of time

y


F = 20N

x


40°




Homework 10D (angular-momentum vector)

7. Find a lamp with a standard incandescent light bulb. With the bulb facing you, unscrew the bulb just a bit. As the bulb is being unscrewed, the angular momentum vector points (a) to the left (b) to the right (c) towards you (d) away from you


8. Screw the bulb back in. As the bulb is being screwed in, the angular momentum vector points (a) to the left (b) to the right (c) towards you (d) away from you
9. For each of the following, sketch the position and velocity vector of the particle in Cartesian three-space, and then compute the angular momentum vector. Use the right-hand rule to check that your result is correct.

(a) A 2.0kg particle with r = (3.0m î + 4.0m ĵ) and v = (2.0m/s î – 1.0m/s ĵ).

(b) A 0.7kg particle with r = (5.0m ĵ + 6.0m k) and v = (4.0m/s k – 8.0m/s ĵ).
10. Shown below is the top view of a merry-go-round, which is rotating counterclockwise at a rate of one revolution every 2.0s. It has a radius R = 1.5m and a mass of 250kg.

(a) What is the angular velocity vector of the merry-go-round?

(b) What is the angular momentum vector of the merry-go-round? (Assume that the merry-go-round is a solid cylinder).

(c) Due to the effect of friction, the merry-go-round comes to rest in 17.0s. What is the torque vector exerted by friction?




y




x


z


11. A string is wrapped around a cylinder which rotates about an axis through its center. The cylinder has mass M = 5kg and radius r = 30cm. A tension of T = 80N is maintained in the string.

(a) What is the torque vector on the cylinder?

(b) What is the resulting angular acceleration vector of the cylinder?

(c) What is the angular velocity vector of the cylinder after the torque has been applied for

t = 5.0s?

(


y
d) What is the angular momentum vector of the cylinder after the torque has been applied for t = 5.0s?


T = 80N







x


Homework 10E (conservation of angular momentum)

12. The sun’s current radius is Ri, its mass is M, and it rotates with angular velocity ωi. Suppose that the sun were to collapse into a neutron star with radius Rf with out any change in its mass.

(a) What would be sun’s final angular velocity? Express your answer in terms of Ri, M, ωi, and Rf.

(b) The sun’s current radius is Ri = 6.96x108 m, and its rotational period is 25.3 earth days. Using this information, and the result of part (a), what would be the numerical value of the sun’s angular velocity if it collapsed into a neutron star with radius 5.0km?

USE TECHNIQUE 6 TO SOLVE THIS PROBLEM
13. A certain lazy Susan is a cylinder with a radius R, and mass M. Initially, it is rotating with an angular velocity ωi. Suddenly, a waiter places a bottle of wine with mass mW at the edge of the lazy Susan.

(a) What is the final angular velocity of the wine-lazy Susan system? Express your answer in terms of R, M, mW, and ωi?

(b) Suppose that R = 1.5m, M = 7.0kg, mW = 0.8kg, and ωi = 2.0rad/s. What is the numerical value of the final angular velocity?

USE TECHNIQUE 6 TO SOLVE THIS PROBLEM


14. Two disks of identical mass but different radii (R and 3R) are spinning on frictionless bearings in the SAME direction. The disk with radius R has angular velocity ω0, and the disk with radius 3R has angular velocity ω0/2. The two disks are brought together slowly. The frictional force between the disks brings them to a common angular velocity ωf. What is ωf in terms of ω0?

USE TECHNIQUE 6 TO SOLVE THIS PROBLEM


Initial Situation, Problem 14




ω0/2


ω0


3R




R

+x


15. A rod with length L and mass M sits on a frictionless surface, and rotates about an axis through its center. A blob of putty with mass m, moving in a direction perpendicular to the center of the rod at speed v, strikes the rod and sticks to it (see diagram below). After the collision, the blob and the rod rotate together at the same angular velocity.

(a) What is the angular velocity of the meterstick-blob combination after the collision? Express your answer in terms of L, M, m, and v.

(b) Suppose that L = 80cm, M = 0.3kg, m = 0.1kg, and v = 2.0m/s. What is the numerical value of the final angular velocity?

USE TECHNIQUE 6 TO SOLVE THIS PROBLEM

M

axis of


rotation





m

v


16. A girl with mass mg is standing a distance b from the center of a merry-go-round with mass M and radius R. A moment later, she catches a cannonball which moves along a tangent to the merry-go-round, as shown below. The cannonball has mass mc, and moves with speed v. What is the angular velocity of the combination of girl, merry-go-round, and cannonball immediately after the cannonball is caught?


x

y

v

cannonball

girl

b
USE TECHNIQUE 6 TO SOLVE THIS PROBLEM


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