Newton's Second Law in Rotational Form
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AP Physics 1 › Newton's Second Law in Rotational Form
A flywheel about a fixed axle has a constant net torque $\tau$ clockwise, producing clockwise angular acceleration $\alpha$. If the flywheel’s rotational inertia is increased by a factor of $5$ while keeping the same $\alpha$, what must happen to $\tau$?
$\tau$ must decrease by a factor of $5$.
$\tau$ becomes proportional to angular velocity, so it increases over time.
$\tau$ is unchanged because torque depends only on force direction.
$\tau$ must increase by a factor of $5$.
Explanation
This problem applies Newton's second law in rotational form to find the required torque when inertia changes. The key equation is τ_net = I·α, which shows that to maintain constant angular acceleration when rotational inertia changes, the torque must change proportionally. If the flywheel's rotational inertia increases by a factor of 5 (from I to 5I) while keeping α constant, then τ_new = (5I)·α = 5(I·α) = 5τ. The torque must increase by a factor of 5 to maintain the same angular acceleration with five times the rotational inertia. Choice A incorrectly suggests decreasing the torque, which would actually reduce the angular acceleration. When solving for required torque, remember that τ and I must change by the same factor to keep α constant.
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 $\alpha$?
$\alpha$ depends on angular velocity, so it cannot be determined.
$\alpha$ stays the same because torque sets angular acceleration directly.
$\alpha$ decreases because less inertia means less rotation.
$\alpha$ increases because $\alpha=\tau/I$.
Explanation
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.
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 $\alpha$?
$\alpha$ is unchanged because torque depends only on force and lever arm.
$\alpha$ decreases because $\alpha=\tau/I$.
$\alpha$ increases because a larger $I$ means more torque is “stored.”
$\alpha$ increases because $\tau$ is proportional to angular velocity.
Explanation
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.
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?
Both have the same $\alpha$ because they have the same net torque.
Both have the same $\alpha$ if they start with the same angular velocity.
Wheel 2 has smaller $\alpha$ because $\alpha=\tau/I$.
Wheel 2 has greater $\alpha$ because it has more mass.
Explanation
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.
A rigid disk rotates about a fixed axis. The net torque is clockwise and constant, and the angular acceleration is clockwise. If both the net torque and rotational inertia are doubled, how does $\alpha$ change?
$\alpha$ becomes half as large.
$\alpha$ becomes four times larger.
$\alpha$ becomes twice as large.
$\alpha$ is unchanged.
Explanation
This question examines Newton's second law in rotational form when both torque and inertia change proportionally. The key relationship is τ_net = I·α, which gives us α = τ_net/I. If both the numerator (torque) and denominator (rotational inertia) are doubled, we get α_new = (2τ)/(2I) = τ/I = α. The angular acceleration remains unchanged because the ratio of torque to inertia stays the same. Choice A incorrectly suggests the acceleration quadruples, perhaps by mistakenly multiplying the two factors of 2. When both τ and I change by the same factor, their ratio (and thus α) remains constant—this is a key insight for analyzing rotational dynamics.
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 $\alpha$?
It becomes zero because the bar resists rotation.
It stays the same.
It doubles.
It becomes half as large.
Explanation
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.
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?
$\alpha_1<\alpha_2$ because $\alpha=\tau/I$.
$\alpha_1>\alpha_2$ because larger inertia means more torque effect.
$\alpha_1=\alpha_2$ because the torques are equal.
Their angular accelerations depend only on their angular velocities.
Explanation
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.
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?
It doubles.
It becomes zero because clockwise torque cancels clockwise acceleration.
It halves.
It is unchanged because $\alpha$ depends on angular velocity.
Explanation
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.
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?
It doubles because $\tau$ is proportional to angular velocity.
It becomes zero because constant torque cannot act at higher speed.
It halves because faster rotation reduces torque.
It is unchanged because $\alpha$ depends on $\tau_{\text{net}}$ and $I$.
Explanation
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.
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?
It becomes zero because lower inertia means less torque needed.
It is unchanged because torque direction is unchanged.
It becomes twice as large.
It becomes half as large.
Explanation
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.