Rotational Kinetic Energy
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AP Physics C: Mechanics › Rotational Kinetic Energy
Based on the scenario, calculate the flywheel’s rotational kinetic energy for $I=0.80\ \text{kg·m}^2$ and $\omega=50\ \text{rad/s}$.
$2000\ \text{N·m}$
$1000\ \text{J}$
$1250\ \text{J}$
$800\ \text{J}$
Explanation
This question tests AP Physics C understanding of rotational kinetic energy in rotating systems. Rotational kinetic energy is calculated using the formula K_rot = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity. In this scenario, we have a flywheel with I = 0.80 kg·m² and ω = 50 rad/s. Choice B is correct because K_rot = (1/2)(0.80)(50)² = (1/2)(0.80)(2500) = 1000 J. Choice D is incorrect because it has the wrong units (N·m instead of J), though numerically it might seem plausible. To help students: Emphasize that rotational kinetic energy has units of joules, not newton-meters (though they are dimensionally equivalent). Practice substituting values carefully and checking units throughout calculations.
Based on the scenario, determine $I$ for $K_\text{rot}=125\ \text{J}$ when $\omega=10\ \text{rad/s}$.
$25\ \text{kg·m}^2$
$1.25\ \text{kg·m}^2$
$2.50\ \text{kg·m}^2$
$0.80\ \text{kg·m}^2$
Explanation
This question tests AP Physics C understanding of rotational kinetic energy in rotating systems. Rotational kinetic energy is calculated using the formula K_rot = (1/2)Iω², which we rearrange to solve for I: I = 2K_rot/ω². In this scenario, we need to find I when K_rot = 125 J and ω = 10 rad/s. Choice C is correct because I = 2(125)/(10)² = 250/100 = 2.50 kg·m². Choice B is incorrect as it represents half the correct value, likely from forgetting to multiply by 2 when rearranging the formula. To help students: When solving for I, remember that the rearranged formula includes a factor of 2 in the numerator. Practice dimensional analysis to verify your algebraic manipulations.
Based on the scenario, if a disk’s angular velocity doubles, what happens to its rotational kinetic energy?
It doubles.
It quadruples.
It halves.
It remains constant.
Explanation
This question tests AP Physics C understanding of rotational kinetic energy in rotating systems. Rotational kinetic energy is calculated using the formula K_rot = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity. In this scenario, we examine what happens when angular velocity doubles while moment of inertia remains constant. Choice C is correct because when ω doubles, ω² becomes four times larger, making K_rot quadruple since K_rot is proportional to ω². Choice B is incorrect because it assumes a linear relationship between K_rot and ω, missing the squared term. To help students: Emphasize the quadratic relationship between rotational kinetic energy and angular velocity. Use graphical representations to show how K_rot varies with ω².
Based on the scenario, calculate $K_\text{rot}$ for a flywheel with $I=2.5\ \text{kg·m}^2$ spinning at $\omega=20\ \text{rad/s}$.
$1000\ \text{J}$
$50\ \text{J}$
$250\ \text{J}$
$500\ \text{J}$
Explanation
This question tests AP Physics C understanding of rotational kinetic energy in rotating systems. Rotational kinetic energy is calculated using the formula K_rot = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity. In this scenario, we calculate K_rot for a flywheel with I = 2.5 kg·m² and ω = 20 rad/s. Choice B is correct because K_rot = (1/2)(2.5)(20)² = (1/2)(2.5)(400) = 500 J. Choice C is incorrect as it omits the factor of 1/2, giving 1000 J instead of 500 J. To help students: Always remember the factor of 1/2 in the rotational kinetic energy formula. Create mnemonics or visual aids to distinguish between formulas that include 1/2 and those that don't.
Using the information above, determine $I$ required for $K_\text{rot}=360\ \text{J}$ at $\omega=30\ \text{rad/s}$.
$0.40\ \text{kg·m}^2$
$24\ \text{kg·m}^2$
$1.20\ \text{kg·m}^2$
$0.80\ \text{kg·m}^2$
Explanation
This question tests AP Physics C understanding of rotational kinetic energy in rotating systems. Rotational kinetic energy is calculated using the formula K_rot = (1/2)Iω², which we rearrange to solve for I: I = 2K_rot/ω². In this scenario, we need to find I when K_rot = 360 J and ω = 30 rad/s. Choice B is correct because I = 2(360)/(30)² = 720/900 = 0.80 kg·m². Choice A is incorrect as it represents half the correct value, possibly from forgetting the factor of 2 when rearranging the formula. To help students: Practice rearranging the rotational kinetic energy formula to solve for different variables. Emphasize careful algebraic manipulation and checking answers by substituting back into the original formula.