Newton's Second Law in Rotational Form - AP Physics 1

1. Two solid cylinders rotate about fixed axles. Cylinder 1 has net counterclockwise torque and counterclockwise angular acceleration. Cylinder 2 has the same net torque but twice the rotational inertia. How does $\alpha_2$ compare to $\alpha_1$?
   Answer: $\alpha_2=\tfrac{1}{2}\alpha_1$ because $\alpha=\tau_{\text{net}}/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, this equation highlights that angular acceleration results from the imbalance of torques, scaled by the object's resistance to rotation via its inertia. When two objects experience the same net torque, the one with greater rotational inertia will have a smaller angular acceleration. Doubling I while keeping τ_net constant therefore halves α, as the torque is distributed over more inertia. A common distractor is choice A, which incorrectly states more inertia leads to more acceleration, confusing inertia with a driving factor rather than resistance. For similar questions, compare scenarios using ratios from α = τ_net / I to quantify changes in angular acceleration.
2. A uniform disk on a frictionless axle has a tangential pull at its rim that produces a net counterclockwise torque; the disk's angular acceleration is counterclockwise. If the same net torque is applied to an otherwise identical disk with larger rotational inertia, what happens to the angular acceleration?
   Answer: It decreases because $\alpha=\tau_{\text{net}}/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, this law means that the net torque applied to an object determines how quickly its angular velocity changes, with the angular acceleration being directly proportional to the torque. The rotational inertia I acts as a measure of resistance to changes in rotational motion, similar to how mass resists linear acceleration. Therefore, for a fixed net torque, a larger rotational inertia results in a smaller angular acceleration, as the same torque has to overcome more resistance. A common distractor is choice A, which incorrectly assumes torque is proportional to angular velocity, but torque actually causes the change in angular velocity regardless of its current value. To approach such problems, always identify what quantities are changing and apply the formula α = τ_net / I to see the effect on angular acceleration.
3. A rotor experiences a net torque counterclockwise about a fixed axis and has angular acceleration counterclockwise. If the net torque is unchanged but the rotational inertia decreases by 20%, what happens to $\alpha$?
   Answer: $\alpha$ increases by 25% because $\alpha=\tau/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, rotational inertia opposes changes in angular velocity, so decreasing it allows greater acceleration for the same torque. The inverse proportionality means a reduction in I increases α by the reciprocal factor. With I decreasing to 80% (0.8 times) and τ unchanged, α increases by 1/0.8 = 1.25 times, or 25%. A common distractor is choice A, claiming α decreases by 20% because inertia decreased, but this mistakes the inverse relationship for a direct one. A key strategy is to express changes as fractions or percentages and use α ∝ 1/I to find the percentage change in α.
4. A turntable experiences a constant net torque $\tau$ counterclockwise, producing counterclockwise angular acceleration $\alpha$. If $\tau$ is reduced to $\tfrac{1}{2}\tau$ while the rotational inertia remains $I$, what is the new angular acceleration?
   Answer: $\tfrac{1}{2}\alpha$ This problem applies Newton's second law in rotational form to find how angular acceleration changes with torque. The rotational equation τ_net = I·α shows that for a given rotational inertia, angular acceleration is directly proportional to net torque. When the net torque is reduced from τ to τ/2 while I remains constant, the angular acceleration becomes α_new = (τ/2)/I = (1/2)(τ/I) = α/2. The angular acceleration is halved because it's directly proportional to the torque when inertia is fixed. Choice D incorrectly claims that any reduction in torque stops angular acceleration entirely, but objects continue to accelerate (just more slowly) as long as there's any net torque. To solve torque-acceleration problems, remember that halving the torque halves the angular acceleration when I is constant.
5. A rigid wheel experiences a net torque clockwise about its center and has angular acceleration clockwise. If the net torque magnitude is doubled while rotational inertia stays constant, how does $\alpha$ change?
   Answer: $\alpha$ doubles because $\alpha\propto\tau$ for constant $I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, torque drives rotational acceleration much like force drives linear acceleration. The magnitude of angular acceleration is directly proportional to the net torque and inversely proportional to rotational inertia, so increasing torque while keeping inertia constant increases acceleration proportionally. Here, doubling the torque with constant inertia results in α doubling, as α = τ / I. A common distractor is choice C, which claims α halves because larger torque reduces acceleration time, but this confuses acceleration with time-dependent quantities like velocity. For similar problems, always isolate the variables in τ = I α and see how changes in τ or I affect α directly.
6. A rigid bar rotates about a frictionless pivot. A constant net torque acts clockwise, and the bar's angular acceleration is clockwise. If the bar's rotational inertia about the pivot is doubled while the net torque is unchanged, what is the new $\alpha$?
   Answer: It becomes half as large. This question applies Newton's second law in rotational form to analyze how angular acceleration changes with rotational inertia. The rotational form states τ_net = I·α, which means the net torque equals the product of rotational inertia and angular acceleration. When net torque is constant, angular acceleration is inversely proportional to rotational inertia: α = τ_net/I. If the rotational inertia doubles while torque remains constant, the angular acceleration becomes half as large because you're dividing the same torque by twice the inertia. Choice D incorrectly suggests the bar would stop rotating, but objects don't "resist" rotation—they simply require more torque to achieve the same angular acceleration when they have more rotational inertia. Remember that doubling I while keeping τ constant always halves α.
7. A rigid disk experiences a net torque clockwise and speeds up clockwise. If the torque is tripled while the disk's rotational inertia is halved, how does the magnitude of $\alpha$ change?
   Answer: It becomes 6 times as large. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, increasing torque amplifies angular acceleration, while decreasing inertia also amplifies it by reducing resistance. The combined effect is multiplicative in the ratio τ / I. Here, tripling τ and halving I results in α becoming 6 times as large, since α' = (3τ) / (I/2) = 6 (τ / I). A common distractor is choice A, suggesting it becomes 3/2 times as large by incorrectly adding factors instead of multiplying. For problems with multiple changes, calculate the overall factor as (new τ / old τ) × (old I / new I) and apply it to the original α.
8. A solid cylinder rotates about its central axis. A constant net torque is applied clockwise, and the angular acceleration is clockwise. The cylinder is replaced by a different object that has smaller rotational inertia about the same axis, while the net torque is unchanged. What happens to $\alpha$?
   Answer: $\alpha$ increases because $\alpha=\tau/I$. This question tests understanding of Newton's second law in rotational form when rotational inertia decreases. The fundamental relationship is τ_net = I·α, which rearranges to α = τ_net/I, showing that angular acceleration is inversely proportional to rotational inertia when torque is constant. When the cylinder is replaced with an object having smaller rotational inertia but the same net torque is applied, the angular acceleration must increase because we're dividing the same torque by a smaller inertia. Choice C incorrectly suggests that torque directly determines angular acceleration without considering inertia, missing the key role that I plays in rotational dynamics. When analyzing rotational motion, always consider both torque and rotational inertia to determine angular acceleration.
9. A uniform bar about a pivot has a net torque directed counterclockwise, producing counterclockwise angular acceleration. Without changing the net torque, the bar is replaced by one with the same shape but larger mass, increasing $I$. What happens to $\alpha$?
   Answer: It decreases. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, net torque provides the rotational 'push' that changes angular velocity, while rotational inertia quantifies how hard it is to make that change. Increasing the mass of an object with the same shape increases its I, making it more resistant to angular acceleration for a given torque. Thus, with τ_net unchanged and I larger, the angular acceleration decreases. A common distractor is choice D, which wrongly suggests increasing I reverses the direction of α, but direction is set by torque, not inertia. Always remember to check if changes affect magnitude or direction separately when applying τ_net = I α.
10. A rigid disk has net torque directed counterclockwise and angular acceleration counterclockwise. If $I$ is tripled and $\tau_{\text{net}}$ is also tripled, what happens to the angular acceleration magnitude?
    Answer: It is unchanged because $\alpha=\tau_{\text{net}}/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, torque and inertia both influence angular acceleration in opposite ways: torque promotes it, inertia resists it. If both τ_net and I are increased by the same factor, their effects cancel out, leaving α unchanged. Tripling both means the increased driving torque is exactly balanced by the increased resistance. A common distractor is choice A, which ignores the inertia change and assumes α triples with torque alone. To tackle these, use proportional reasoning: if τ scales by k and I by k, α remains the same.
11. A wheel has net torque counterclockwise and angular acceleration counterclockwise. Two wheels experience the same net torque, but wheel 2 has greater rotational inertia. Which statement is correct?
    Answer: Wheel 2 has smaller $\alpha$ because $\alpha=\tau/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, for the same torque, an object with greater rotational inertia will experience smaller angular acceleration due to increased resistance to rotational change. The relationship is inverse: α decreases as I increases. Thus, wheel 2 with greater I has smaller α than wheel 1 under the same τ. A common distractor is choice C, stating both have the same α because of the same net torque, but this overlooks inertia's role. When comparing accelerations in similar scenarios, always account for differences in I by using α = τ / I for each object.
12. A wheel about a fixed axle has net torque clockwise and angular acceleration clockwise. For the same wheel, the angular velocity is doubled but the net torque is unchanged. What happens to the angular acceleration?
    Answer: It is unchanged because $\alpha$ depends on $\tau_{\text{net}}$ and $I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, this law indicates that angular acceleration depends solely on current net torque and rotational inertia, not on the existing angular velocity. Changing the angular velocity without altering τ_net or I leaves α unaffected. The acceleration remains constant as long as τ_net and I are fixed, regardless of speed. A common distractor is choice A, which confuses torque with being proportional to velocity, but torque is independent of ω in this context. When analyzing rotational motion, separate kinematic variables like ω from dynamic ones like τ and I.
13. A rigid rod about a fixed pivot experiences a net clockwise torque and has a clockwise angular acceleration. If the net torque is doubled while the rod's rotational inertia is unchanged, how does $\alpha$ change?
    Answer: It doubles. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, this relationship shows that angular acceleration is the result of net torque divided by rotational inertia, meaning torque drives the change in rotational motion. When rotational inertia is constant, increasing the net torque directly increases the angular acceleration in proportion. This is analogous to linear motion where doubling the force doubles the acceleration for a fixed mass. A common distractor is choice C, which wrongly suggests angular acceleration depends on angular velocity, but α is independent of current ω and depends only on τ_net and I. To solve these problems effectively, isolate the changing variable in τ_net = I α and determine its proportional impact on α.
14. A flywheel experiences a net clockwise torque and thus a clockwise angular acceleration. If the flywheel's rotational inertia is reduced to half while the same net torque is applied, how does the angular acceleration magnitude change?
    Answer: It becomes twice as large. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, this means angular acceleration is produced by net torque and opposed by the object's rotational inertia. Reducing rotational inertia makes the object easier to accelerate rotationally, so for the same torque, α increases. Specifically, halving I while keeping τ_net constant doubles the angular acceleration. A common distractor is choice A, which reverses the relationship and claims α halves, misunderstanding inertia's inverse role. A useful strategy is to rearrange τ_net = I α to solve for α and plug in the modified values to see the outcome.
15. A uniform disk spins about a fixed axle. A tangential force produces a constant net torque $\tau$ counterclockwise, and the disk's angular acceleration is counterclockwise. If $\tau$ stays the same but the disk is replaced by one with larger rotational inertia $I$, what happens to $\alpha$?
    Answer: $\alpha$ decreases because $\alpha=\tau/I$. This question tests Newton's second law in rotational form. Newton's second law for rotation states that the net torque equals the rotational inertia times the angular acceleration: τ_net = I·α. When a constant net torque acts on a rotating object, the angular acceleration depends inversely on the rotational inertia—objects with larger I are harder to accelerate rotationally, just as objects with larger mass are harder to accelerate linearly. Since α = τ/I, if τ remains constant but I increases, then α must decrease proportionally. Choice A incorrectly suggests that torque can be "stored" in rotational inertia, which confuses the concepts. To solve problems like this, identify what quantities remain constant (here, τ) and apply the rotational form of Newton's second law algebraically.
16. A solid cylinder and a hoop (same mass and radius) each experience the same net counterclockwise torque and accelerate counterclockwise. Which has the greater angular acceleration?
    Answer: The solid cylinder, because it has smaller rotational inertia. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, objects with mass distributed closer to the axis of rotation have smaller rotational inertia, making them easier to accelerate rotationally. Angular acceleration is inversely proportional to I for a given torque, so a smaller I leads to larger α. In this case, the solid cylinder has smaller I than the hoop for the same mass and radius, resulting in greater α for the cylinder under the same torque. A common distractor is choice A, suggesting the hoop has larger torque due to mass distribution, but torque is externally applied and the same for both. To tackle similar questions, calculate or compare rotational inertias first, then apply α = τ / I to compare accelerations.
17. A rigid wheel experiences a constant net torque counterclockwise, and its angular acceleration is counterclockwise. Two wheels experience the same net torque magnitude, but wheel 1 has larger rotational inertia than wheel 2. Which statement about their angular accelerations is correct?
    Answer: $\alpha_1<\alpha_2$ because $\alpha=\tau/I$. This question compares angular accelerations using Newton's second law in rotational form. The fundamental relationship τ_net = I·α rearranges to α = τ_net/I, showing that angular acceleration is inversely proportional to rotational inertia when torque is constant. Since both wheels experience the same net torque magnitude but wheel 1 has larger rotational inertia than wheel 2, we can write α₁ = τ/I₁ and α₂ = τ/I₂. Because I₁ > I₂ and the torques are equal, we must have α₁ < α₂—the wheel with larger inertia has smaller angular acceleration. Choice A incorrectly reverses this relationship, suggesting larger inertia leads to greater angular acceleration. To compare angular accelerations, always check which object has larger I: the one with larger I will have smaller α for the same torque.
18. A uniform rod about a fixed pivot has a constant net torque in the clockwise direction and thus a clockwise angular acceleration. If the net torque is tripled while the rod's rotational inertia about the pivot is unchanged, what happens to $\alpha$?
    Answer: $\alpha$ is tripled. This question tests Newton's second law in rotational form when net torque changes. The rotational form states τ_net = I·α, which means angular acceleration is directly proportional to net torque when rotational inertia is constant. If the net torque is tripled from τ to 3τ while the rod's rotational inertia remains unchanged, then α_new = (3τ)/I = 3(τ/I) = 3α. The angular acceleration triples because it's directly proportional to the applied torque. Choice C incorrectly suggests that larger torque increases resistance, confusing cause and effect—torque causes angular acceleration, not resistance to it. To analyze rotational problems, remember that tripling the torque triples the angular acceleration when I is constant.
19. A uniform disk initially at rest experiences a net torque $\tau$ counterclockwise about its axle and begins accelerating counterclockwise with angular acceleration $\alpha$. If the same torque is applied to an otherwise identical disk with twice the rotational inertia, what happens to $\alpha$?
    Answer: $\alpha$ is halved because $\alpha=\tau/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, torque is the rotational equivalent of force, causing a change in rotational motion. The angular acceleration produced is directly proportional to the net torque applied and inversely proportional to the object's rotational inertia, meaning that for a given torque, a larger inertia results in smaller acceleration. In this scenario, since the torque remains the same but the rotational inertia doubles, the angular acceleration must be halved to maintain the relationship τ = I α. A common distractor is choice B, which suggests α is unchanged because torque is unchanged, but this ignores the inverse relationship with inertia. When approaching similar problems, identify how torque and inertia change and apply α = τ_net / I to determine the effect on angular acceleration.
20. A rigid rotor experiences a net torque clockwise and has angular acceleration clockwise. If both the net torque and the rotational inertia are doubled, how does $\alpha$ change?
    Answer: $\alpha$ is unchanged because $\alpha=\tau/I$. This question assesses understanding of Newton's second law for rotation, which states that net torque equals rotational inertia times angular acceleration, τ_net = I α. Qualitatively, torque and inertia both influence angular acceleration in opposing ways: more torque increases it, more inertia decreases it. When both increase by the same factor, their effects cancel out, leaving α unchanged. In this scenario, doubling both τ and I results in α remaining the same, since α = (2τ) / (2I) = τ / I. A common distractor is choice A, focusing only on torque doubling to say α doubles, but this neglects the simultaneous increase in inertia. For analogous questions, consider the ratio of changes in τ and I, as α scales with τ / I.

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