If a solid disk rolls without slipping, what fraction of its total energy is tra

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Question: If a solid disk rolls without slipping, what fraction of its total energy is translational at the bottom of an incline?

Options:

Correct Answer: 1/3

Solution:

For a solid disk, the translational kinetic energy is 1/3 of the total kinetic energy when rolling without slipping.

If a solid disk rolls without slipping, what fraction of its total energy is translational at the bottom of an incline?

If a solid disk rolls without slipping, what fraction of its total energy is translational at the bottom of an incline?

Step 1: Understand that a solid disk has two types of kinetic energy when it rolls: translational kinetic energy and rotational kinetic energy.
Step 2: Recall the formula for translational kinetic energy, which is (1/2)mv^2, where m is mass and v is velocity.
Step 3: Recall the formula for rotational kinetic energy, which is (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 4: For a solid disk, the moment of inertia I is (1/2)mr^2, where r is the radius of the disk.
Step 5: When the disk rolls without slipping, the relationship between linear velocity v and angular velocity ω is v = rω.
Step 6: Substitute ω in the rotational kinetic energy formula using v = rω, which gives you (1/2)(1/2)mr^2(v/r)^2 = (1/4)mv^2.
Step 7: Now, add the translational and rotational kinetic energies to find the total kinetic energy: Total KE = (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.
Step 8: To find the fraction of the total energy that is translational, divide the translational kinetic energy by the total kinetic energy: (1/2)mv^2 / (3/4)mv^2.
Step 9: Simplify the fraction: (1/2) / (3/4) = (1/2) * (4/3) = 2/3.
Step 10: Conclude that the fraction of the total energy that is translational at the bottom of the incline is 2/3.

Rolling Motion – Understanding the relationship between translational and rotational kinetic energy in rolling objects.
Energy Distribution – Analyzing how total energy is divided between translational and rotational forms in a rolling disk.