phy191 exam
Rolling Motion & Angular Momentum
4/2/2108
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Table Challenge 5
A pencil has mass M and length L. It is standing straight up on a table on its eraser end. A slight push causes the pencil to fall over. Friction between the eraser and the table provides a pivot point.
Find an expression for the speed of the tip of the pencil as it hits the table.
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Mass M
Length L
Pivot
Point
Rolling motion without slipping is difficult to describe with kinematics
So instead, use energy
The text describes rolling as:
Rotational kinetic energy can be added in with translations to find the total kinetic energy of an object:
Rolling Motion
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Gotta Go Fast
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We’ll be taking a look at the rolling motion of 4 different objects:
Solid Sphere, Hollow Sphere, Solid Cylinder, Hollow Cylinder
Using the materials provided, set up an incline plane and rank the speeds of the objects from fastest (shortest time) to slowest (longest time)
When finished, record your ranking on LC
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An Analysis of Motion
There are several ways you can find how long it takes an object to slide down an incline without friction…
What about for rolling motion?
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0
1
L
y = 0
From 1D Kinematics for x-axis parallel to incline:
Conserve energy 0 to 1:
Whiteboard Problem 12-14
Find an expression for the time it takes for a solid sphere to go down an incline with no rolling friction. Assume that the object is rotating about its center of mass, so that the moment of inertia can be found in a table. Your expression should look like:
Enter your number in (LC)
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Whiteboard Problem 12-5
A marble starts from rest and rolls down the track shown around the loop-the-loop of radius R. The marble has mass m and radius r.
Find an expression for the minimum height h that the track must have for the marble to make it around the loop without falling off. (LC)
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Angular Momentum
When an object rotates, it has rotational kinetic energy
It will also have angular momentum.
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For a particle:
Uniform Circular Motion:
Rigid Body rotating about a fixed symmetry axis:
The vectors in the rigid body momentum just indicate direction as ccw or cw
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Conservation of Angular Momentum
Recall that Newton’s second law could be written in terms of momentum
So then for a rotating system:
Which if there are no net torques…
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In Summary
Kinematics:
Dynamics:
Energy:
Momentum:
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Constant Acceleration
(particle)
(rigid body)
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Whiteboard Problem 12-16
A 2.0kg, 20cm diameter turntable rotates at 100 rpm on frictionless bearings. Two 500g blocks fall from above, hit the turntable simultaneously at the opposite end of a diameter, and stick.
What is the turntable’s angular velocity, in rpm, after this event? (LC)
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