Ring, Disc, and Sphere, 1Q10.30

Topic and Concept:

• Rotational Dynamics, 1Q10. Moment of Inertia

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Location:

• ME South Wall

Abstract:

A ring, disc, sphere, and weighted discs of the same diameter are rolled down an incline. Each object reaches the bottom at different times due to differing moments of inertia.

Important Setup Notes:

• N/A

Setup and Procedure:

1. Unload the ring, disc, and spheres from the cart.
2. Position the cart at the stage right end of the lecture bench so that it is touching the bench (see pictures).
3. Place the ramp on top of the lecture bench so that it is in line with the cart and such that the bottom edge of the ramp is within about 2 inches of the cart.
4. Secure the bottom of the ramp to the bench using the C-clamp provided in the cart.
5. Use the provided weights to hold down the top end of the bench (see pictures).

6. Either roll each object down individually or in pairs to directly compare descent times. A stop watch can be used to quantitatively measure the times.

Cautions, Warnings, or Safety Concerns:

• N/A

Discussion:

For a system of N particles, the moment of inertia can be defined as I = Σ_im_i*r_i² where i ranges from 1 to N. This is important when considering the rotational dynamics of a system. In our case, we are dealing with rigid bodies and can consider the mass distribution to be continuous and uniform in each object consisting of one material. In this case we can replace the discrete sum with an integral over the volume of the object: I = ρ*∫ r² dV. This quantity is analogous to mass in translational motion. In our case, the moments of inertia for these three objects are

• Hoop: I = M*R²

• Disc: I = (1/2)*M*R²

• Sphere: I = (2/5)*M*R²

where M is the total mass, R is the radius, and the axes of rotation for the hoop and disc is taken to be the axis of rotational symmetry. Each object begins to roll down the ramp (i.e. attains an angular acceleration) because there is a net torque on the objects, the acceleration being determined by α = τ_net/I. This shows us that the greater the moment of inertia of an object, the smaller its angular acceleration will be. This is why each object reaches the bottom of the ramp at different times.

Videos:

• Lecture Demonstration's Youtube Channel

References:

• Wikipedia - Moment of Inertia

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