Flashcards · AP® Physics 1: Algebra-Based · Unit 5

Unit 5: Torque and Rotational Dynamics flashcards

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5.1 Rotational Kinematics

What quantities describe rotation about a fixed axis?: Angular displacement Δθ (rad), angular velocity ω_avg = Δθ / Δt (rad/s) and angular acceleration α_avg = Δω / Δt (rad/s²), with a stated positive sense.
Do all points on a rigid rotating body have the same angular velocity?: Yes. All points share the same Δθ, ω and α. Only linear speed and tangential acceleration change with distance from the axis.
What are the rotational kinematic equations?: For constant α: ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2α(θ − θ₀).
How do you convert revolutions and rpm into radians?: 1 rev = 2π rad = 360°. For rpm, multiply by 2π and divide by 60: 45 rpm ≈ 4.7 rad/s. Always convert before using the equations.
How do you tell if a rotating object is speeding up or slowing down?: Compare the signs of ω and α. Same signs: speeding up. Opposite signs: slowing down.
What do rotational graphs show?: Slope of θ–t = ω. Slope of ω–t = α. Area under ω–t = angular displacement.

5.2 Connecting Linear and Rotational Motion

How are linear and angular quantities linked for a point at radius r?: Δs = rΔθ, v = rω and a_T = rα, with angles in radians.
What is the centripetal acceleration of a point on a rotating body?: a_c = v²/r = ω²r. On one body (same ω), points farther out have more centripetal acceleration.
Two pulleys are joined by a belt that does not slip. What do they share?: The same rim speed v, not the same ω: r₁ω₁ = r₂ω₂. The smaller pulley turns faster.
A disc spins at 12 rad/s. What is the speed of a point 0.25 m from the axis?: v = rω = 0.25 × 12 = 3.0 m/s.
Is a_T = rα the total acceleration of a point on a spinning body?: No. It changes the speed. Add the perpendicular centripetal part ω²r, which changes the direction.

5.3 Torque

What is torque and how is it calculated?: How strongly a force tends to turn an object about an axis: τ = rF sin θ, with θ between r and F. Equivalent forms: τ = rF⊥ = r⊥F. Unit: N·m (not J).
What is the lever arm?: The perpendicular distance from the axis to the force's line of action, r⊥ = r sin θ. Then τ = r⊥F.
When does a force exert zero torque?: When it acts at the axis, or its line of action passes through the axis, so the lever arm is zero.
A 30 N force acts 0.50 m from the axis at 30° to r. What is the torque?: τ = rF sin θ = 0.50 × 30 × sin 30° = 7.5 N·m.
How is a force diagram for torque different from a free-body diagram?: Each force is drawn at the point where it acts, not from a single dot. Weight acts at the centre of mass (the middle of a uniform rod).
Why must you state the axis for torque?: The same force has different torques about different axes. Label each torque clockwise or counterclockwise about that axis.

5.4 Rotational Inertia

What is rotational inertia?: I measures how hard it is to change a rigid system's rotation. For point masses I = Σmᵢrᵢ², in kg·m², with each r the perpendicular distance from the same axis.
A small mass moves from 0.15 m to 0.30 m from the axis. How does its rotational inertia change?: It becomes 4 times as large, because I ∝ r².
What is the parallel axis theorem?: I = I_cm + Md², where I_cm is about an axis through the center of mass, M is the total mass and d the distance between the parallel axes.
About which axis is rotational inertia smallest?: For a family of parallel axes, the one through the center of mass.
Which has more rotational inertia: a hoop or a solid disk of the same mass and radius?: The hoop, because all its mass is at the rim, far from the axis. Mass distribution matters, not just total mass.
Does an object have one rotational inertia?: No. It has a different I for every axis, so always state the axis.

5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form

What are the conditions for rotational equilibrium?: Στ = 0 about the axis. A rigid system then keeps a constant angular velocity, which includes being at rest or spinning steadily.
Where should you put the pivot when solving a static equilibrium problem?: At a point where an unknown force acts, so that force drops out of the torque equation. For an object in full equilibrium, net torque is zero about any point.
A 2.0 kg mass hangs 0.60 m left of a light see-saw's pivot. What mass 0.40 m to the right balances it?: Torques balance: 2.0 × 0.60 = m × 0.40, so m = 3.0 kg.
Can a system have zero net force but a nonzero net torque?: Yes. A couple (two equal, opposite forces on different lines) gives ΣF = 0 but Στ ≠ 0, so it starts the object turning.
A uniform 6.0 kg plank rests on supports at its two ends. What force does each support exert?: By symmetry each takes half the weight: ½ × 6.0 × 9.8 ≈ 29 N. A uniform beam's weight acts at its middle.
How do you show balanced or unbalanced torques on an ω–t graph?: A flat ω–t graph means Στ = 0. A sloping graph means the torques are unbalanced.

5.6 Newton’s Second Law in Rotational Form

What is Newton's second law in rotational form?: α = τ_net / I, or Στ = Iα. α is proportional to net torque, inversely proportional to rotational inertia, and has the same sense as the net torque.
A motor exerts 5.0 N·m on a rotor with I = 0.50 kg·m², and friction exerts 1.0 N·m the other way. What is α?: τ_net = 5.0 − 1.0 = 4.0 N·m, so α = 4.0 / 0.50 = 8.0 rad/s².
Does angular acceleration point in the direction of angular velocity?: Not necessarily: it follows the net torque. A clockwise wheel with a counterclockwise net torque slows down.
Is the tension in a rope holding a mass on a massive pulley equal to the mass's weight?: No. When the hanging mass accelerates down, T is less than mg. The tensions on the two sides of a massive pulley also differ; their difference × R equals Iα.
How are the linear and rotational motion of a rope and pulley linked?: For a rope that does not slip, a = Rα. Apply ΣF = ma to the hanging object and Στ = Iα to the pulley, with matching sign conventions.
The net torque on a body is tripled and its rotational inertia doubled. How does α change?: α = τ/I changes by a factor of 3/2 = 1.5.

Study Unit 5 further

5.1 Rotational Kinematics Study guide: 5.1 Rotational Kinematics Revision notes: 5.1 Rotational Kinematics Practice questions: 5.1 Rotational Kinematics
5.2 Connecting Linear and Rotational Motion Study guide: 5.2 Connecting Linear and Rotational Motion Revision notes: 5.2 Connecting Linear and Rotational Motion Practice questions: 5.2 Connecting Linear and Rotational Motion
5.3 Torque Study guide: 5.3 Torque Revision notes: 5.3 Torque Practice questions: 5.3 Torque
5.4 Rotational Inertia Study guide: 5.4 Rotational Inertia Revision notes: 5.4 Rotational Inertia Practice questions: 5.4 Rotational Inertia
5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form Study guide: 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form Revision notes: 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form Practice questions: 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form
5.6 Newton’s Second Law in Rotational Form Study guide: 5.6 Newton’s Second Law in Rotational Form Revision notes: 5.6 Newton’s Second Law in Rotational Form Practice questions: 5.6 Newton’s Second Law in Rotational Form

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