Week 6: Vector Torque and Angular Momentum |
305 |
Homework for Week 6
Problem 1.
Physics Concepts: Make this week’s physics concepts summary as you work all of the problems in this week’s assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were key to, and include concepts from previous weeks as necessary. Do the work carefully enough that you can (after it has been handed in and graded) punch it and add it to a three ring binder for review and study come finals!
Problem 2.
ω f
M
R |
v |
m
This problem will help you learn required concepts such as:
•Angular Momentum Conservation
•Inelastic Collisions
so please review them before you begin.
Satchmo, a dog with mass m runs and jumps onto the edge of a merry-go-round (that is initially at rest) and then sits down there to take a ride as it spins. The merry-go-round can be thought of as a disk of radius R and mass M , and has approximately frictionless bearings in its axle. At the time of this angular “collision” Satchmo is travelling at speed v perpendicular to the radius of the merry-go-round and you can neglect Satchmo’s moment of inertia about an axis through his OWN center of mass compared to that of Satchmo travelling around the merry-go-round axis (because R of the merry-go-round is much larger than Satchmo’s size, so we can treat him like “a particle”).
a)What is the angular velocity ωf of the merry-go-round (and Satchmo) right after the collision?
b)How much of Satchmo’s initial kinetic energy was lost, compared to the final kinetic energy of Satchmo plus the merry-go-round, in the collision? (Believe me, no matter what he’s lost it isn’t enough – he’s a border collie and he has plenty more!)
306 |
Week 6: Vector Torque and Angular Momentum |
Problem 3.
ωp
M.R |
ω |
θ |
D |
This problem will help you learn required concepts such as:
•Vector Torque
•Vector Angular Momentum
•Geometry of Precession
so please review them before you begin.
A top is made of a disk of radius R and mass M with a very thin, light nail (r R and m M ) for a spindle so that the disk is a distance D from the tip. The top is spun with a large angular velocity ω. When the top is spinning at a small angle θ with the vertical (as shown) what is the angular frequency ωp of the top’s precession? Note that this is a required problem that will be on at least one test in one form or another, so be sure that you have mastered it when you are done with the homework!
Week 6: Vector Torque and Angular Momentum |
307 |
Problem 4.
d
L
m
v
M
This problem will help you learn required concepts such as:
•Angular Momentum Conservation
•Momentum Conservation
•Inelastic Collisions
•Impulse
so please review them before you begin. You may also find it useful to read: Wikipedia: http://www.wikipedia
A rod of mass M and length L rests on a frictionless table and is pivoted on a frictionless nail at one end as shown. A blob of putty of mass m approaches with velocity v from the left and strikes the rod a distance d from the end as shown, sticking to the rod.
•Find the angular velocity ω of the system about the nail after the collision.
•Is the linear momentum of the rod/blob system conserved in this collision for a general value of d? If not, why not?
•Is there a value of d for which it is conserved? If there were such a value, it would be called the center of percussion for the rod for this sort of collision.
All answers should be in terms of M , m, L, v and d as needed. Note well that you should clearly indicate what physical principles you are using to solve this problem at the beginning of the work.
308 |
Week 6: Vector Torque and Angular Momentum |
Problem 5.
mr
v 
F
This problem will help you learn required concepts such as:
•Torque Due to Radial Forces
•Angular Momentum Conservation
•Centripetal Acceleration
•Work and Kinetic Energy
so please review them before you begin.
A particle of mass m is tied to a string that passes through a hole in a frictionless table and held. The mass is given a push so that it moves in a circle of radius r at speed v.
a)What is the torque exerted on the particle by the string?
b)What is the magnitude of the angular momentum L of the particle in the direction of the axis of rotation (as a function of m, r and v)?
c)Show that the magnitude of the force (the tension in the string) that must be exerted to keep the particle moving in a circle is:
L2
F= mr3
d)Show that the kinetic energy of the particle in terms of its angular momentum is:
K =
L2
2mr2
Now, suppose that the radius of the orbit and initial speed are ri and vi, respectively. From under the table, the string is slowly pulled down (so that the puck is always moving in an approximately circular trajectory and the tension in the string remains radial) to where the particle is moving in a circle of radius r2.
e)Find its velocity v2 using angular momentum conservation. This should be very easy.
f)Compute the work done by the force from part c) above and identify the answer as the workkinetic energy theorem.
Week 6: Vector Torque and Angular Momentum |
309 |
Problem 6.
z
vin
m
r
θ
pivot |
y |
x
This problem will help you learn required concepts such as:
• Vector Torque
~
• Non-equivalence of L and Iω~.
so please review them before you begin.
In the figure above, a mass m is being spun in a circular path at a constant speed v around the z-axis on the end of a massless rigid rod of length r pivoted at the orgin (that itself is being pushed by forces not shown). All answers should be given in terms of m, r, θ and v. Ignore gravity and friction.
a)Find the angular momentum vector of this system relative to the pivot at the instant shown and draw it in on a copy of the figure.
b)Find the angular velocity vector of the mass m at the instant shown and draw it in on the
~ |
2 |
sin |
2 |
(θ) is the moment of inertia around the z axis? |
figure above. Is L = Iz ω~, where Iz = mr |
|
|
What component of the angular momentum is equal to Iz ω?
c)What is the magnitude and direction of the torque exerted by the rod on the mass (relative to the pivot shown) in order to keep it moving in this circle at constant speed? (Hint: Consider the precession of the angular momentum around the z-axis where the precession frequency is ω, and follow the method presented in the textbook above.)
d)What is the direction of the force exerted by the rod in order to create this torque? (Hint: Shift gears and think about the actual trajectory of the particle. What must the direction of the force be?)
e)Set τ = rF or τ = r F and determine the magnitude of this torque. Note that it is equal in magnitude to your answer to c) above and has just the right direction!
312 |
Week 7: Statics |