If a solid cylinder rolls without slipping, what fraction of its total kinetic e

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Question: If a solid cylinder rolls without slipping, what fraction of its total kinetic energy is translational?

Options:

Correct Answer: 2/3

Solution:

For a solid cylinder, the total kinetic energy is KE_total = KE_translational + KE_rotational = (1/2)mv^2 + (1/2)(1/2)mR^2(ω^2). Since ω = v/R, the translational part is 2/3 of the total.

If a solid cylinder rolls without slipping, what fraction of its total kinetic e

Practice Questions

If a solid cylinder rolls without slipping, what fraction of its total kinetic energy is translational?

Questions & Step-by-Step Solutions

If a solid cylinder rolls without slipping, what fraction of its total kinetic energy is translational?

Step 1: Understand that a solid cylinder has two types of kinetic energy: translational and rotational.
Step 2: The formula for total kinetic energy (KE_total) is KE_total = KE_translational + KE_rotational.
Step 3: The translational kinetic energy (KE_translational) is given by the formula (1/2)mv^2, where m is mass and v is velocity.
Step 4: The rotational kinetic energy (KE_rotational) for a solid cylinder is (1/2)(1/2)mR^2(ω^2), where R is the radius and ω is the angular velocity.
Step 5: Since the cylinder rolls without slipping, we can relate angular velocity (ω) to linear velocity (v) using the formula ω = v/R.
Step 6: Substitute ω = v/R into the rotational kinetic energy formula to get KE_rotational = (1/2)(1/2)mR^2(v/R)^2.
Step 7: Simplify the rotational kinetic energy formula to get KE_rotational = (1/4)mv^2.
Step 8: Now, substitute KE_translational and KE_rotational back into the total kinetic energy formula: KE_total = (1/2)mv^2 + (1/4)mv^2.
Step 9: Combine the terms: KE_total = (2/4)mv^2 + (1/4)mv^2 = (3/4)mv^2.
Step 10: To find the fraction of the total kinetic energy that is translational, divide KE_translational by KE_total: (KE_translational / KE_total) = ((1/2)mv^2) / ((3/4)mv^2).
Step 11: Simplify the fraction: (1/2) / (3/4) = (1/2) * (4/3) = 2/3.
Step 12: Conclude that the fraction of the total kinetic energy that is translational is 2/3.

Kinetic Energy of Rolling Objects – Understanding the distribution of kinetic energy in rolling objects, specifically how it is divided into translational and rotational components.
Moment of Inertia – Knowledge of the moment of inertia for a solid cylinder and its role in calculating rotational kinetic energy.
Rolling Without Slipping – The condition that relates translational velocity and angular velocity for rolling objects.

If a solid cylinder rolls without slipping, what fraction of its total kinetic e

What’s inside this PDF?

If a solid cylinder rolls without slipping, what fraction of its total kinetic e

Practice Questions

Questions & Step-by-Step Solutions

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