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Angular Momentum - Physics for Scientists and Engineers I - Solved Problem Sets, Exercises of Engineering Physics

Alliance UniversityEngineering Physics

The solved problems for Physics for Scientists and Engineers course. These problems are toughest at their peak. See the solution and enjoy. Some points are: Angular Momentum, Precession, Student on Stool, External Torques, Angular Momentum, Kinetic Energy, Frictionless, Gravity, Radius and Angular Velocity, Torque

Typology: Exercises

2012/2013

Uploaded on 09/27/2013

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PHYS 2210 — Spring 2013
GA19 Solutions
Problem 1 Angular Momentum: Spinning Disks
Two disks are mounted on frictionless bearings on the same axle and can be brought together so
that they couple and rotate as one unit. The first disk, which has a mass of 1 kg and radius of 2 m,
is set spinning at 45 rad/s. The second disk, which has a mass of 2 kg and a radius of 3 m, is set
spinning at 25 rad/s in the opposite direction. They then couple together. What is their angular
speed after coupling?
Solution:
!1
!2
!f
In the collision of the two spinning disks, angular momentum is conserved. Angular momentum
is the product of the moment of inertia and angular velocity. The angular momentum of the pair
before they collide is the sum of their individual angular momentums; however, remember that they
are spinning in opposite directions and so their angular momentums’ have opposite signs. After
they collide the angular momentum is the product of their combined moment of inertia and the
new angular speed. Solving this equation you should find the new angular speed is 12.3 rad/s.
Problem 2 Angular Momentum: Dumbbell Col lision
A dumbbell consists of two balls, one of mass Mand the other of mass 2M, connected by a light
rod of length L. The dumbbell is mounted vertically on a pivot with the heavier ball at the bottom.
The pivot is located at the midpoint of the rod. The system, which is initially at rest, is free to
rotate about the pivot. A wad of putty of mass Mand initial velocity Vcollides with and sticks to
the lower mass, as shown in the diagram. In terms of the quantities given, what is the minimum
value of Vfor which the dumbbell will make it all the way around?
1
pf3
pf4

Partial preview of the text

Download Angular Momentum - Physics for Scientists and Engineers I - Solved Problem Sets and more Exercises Engineering Physics in PDF only on Docsity!

PHYS 2210 — Spring 2013

GA19 Solutions

Problem 1 Angular Momentum: Spinning Disks

Two disks are mounted on frictionless bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, which has a mass of 1 kg and radius of 2 m, is set spinning at 45 rad/s. The second disk, which has a mass of 2 kg and a radius of 3 m, is set spinning at 25 rad/s in the opposite direction. They then couple together. What is their angular speed after coupling?

Solution:

! 2 !f

In the collision of the two spinning disks, angular momentum is conserved. Angular momentum is the product of the moment of inertia and angular velocity. The angular momentum of the pair before they collide is the sum of their individual angular momentums; however, remember that they are spinning in opposite directions and so their angular momentums’ have opposite signs. After they collide the angular momentum is the product of their combined moment of inertia and the new angular speed. Solving this equation you should find the new angular speed is 12.3 rad/s.

Problem 2 Angular Momentum: Dumbbell Collision

A dumbbell consists of two balls, one of mass M and the other of mass 2M , connected by a light rod of length L. The dumbbell is mounted vertically on a pivot with the heavier ball at the bottom. The pivot is located at the midpoint of the rod. The system, which is initially at rest, is free to rotate about the pivot. A wad of putty of mass M and initial velocity V collides with and sticks to the lower mass, as shown in the diagram. In terms of the quantities given, what is the minimum value of V for which the dumbbell will make it all the way around?

Solution:

!

During the collision, the angular momentum is conserved; therefore, after the collision, the dumbbell will have an angular velocity. It must have enough angular velocity to get all the way around. What will stop it from reaching the top? Work done by gravity. Use the conservation of angular momentum to find the angular velocity after the collision. Then find the rotational kinetic energy of the dumbbell when it first begins to rotate. The rotational energy of the dumbbell plus the gravitational potential energy of the top of the dumbbell must equal the gravitational potential energy of the collided masses when they are in the top position for the dumbbell to be able to make it all the way around. Solving the conservation of angular momentum equation and the conservation of energy equation, you will find a needed initial velocity of:

v =

16 gL. (1)

Problem 3 Statics & Dynamics: Falling Meter Stick

A uniform meter stick of mass 1.5 kg is attached to the wall by a frictionless hinge at one end. On the opposite end it is supported by a vertical massless string such that the stick makes an angle of 40 ◦^ with the horizontal.

  1. Find the tension in the string and the magnitude and direction of the force exerted on the stick by the hinge.
  2. Suppose the string is cut. Find the angular acceleration of the stick immediately thereafter.

Solution

Part 1:

  • We will apply Newton’s 2nd Law for translation and rotation, where we know that the linear and angular accelerations are both zero.
  • Take the hinge as the pivot point for the torque to keep the unknown hinge force from appearing in the rotational equation.
  • Now the math:

0 +

L

cos θ =

M L^2

α (8)

=⇒ α =

g L

cos θ (9)

9 .81 m/s^2 1 m

cos 40◦^ = 11.27 rad/s^2. (10)