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Flashcards · AP® Physics C: Mechanics · Unit 6

Unit 6: Energy and Momentum of Rotating Systems flashcards

46 cards on topics 6.1 to 6.6. Pick a topic or study the whole unit.

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6.1 Rotational Kinetic Energy

What is rotational kinetic energy?
K = ½Iω² for a rigid body turning about a fixed axis, with I about that axis and ω in rad/s. It comes from adding ½v² dm with v = rω.
What is the kinetic energy of a body that moves and spins?
K = ½Mv_cm² + ½I_cm ω²: translation of the center of mass plus rotation about the center of mass. I must be about the center of mass in this form.
What is the double-counting mistake with rotational kinetic energy?
Writing ½I_pivot ω² + ½Mv_cm². ½I_pivot ω² about a fixed pivot is already the total; the split formula uses I_cm.
Can a body have kinetic energy when its center of mass is at rest?
Yes. A flywheel on a fixed axle has zero momentum but kinetic energy ½Iω².
Can rotational kinetic energy be negative or cancel?
No. It is a scalar and never negative. Two parts spinning in opposite senses have kinetic energies that add, not cancel.
How do you find kinetic energy for a non-uniform rotating body?
Start from K = ∫½v² dm with v = rω, which gives ½ω²∫r² dm. Integrate with λ(x) rather than using uniform-rod formulas.
How does rotational kinetic energy grow under constant α from rest?
K = ½Iα²t² = Iαθ: a parabola against time and a straight line against angle.

6.2 Torque and Work

How much work does a torque do?
W = ∫τ dθ with θ in radians, the signed area under the τ–θ graph. For a constant torque, W = τΔθ.
When does a torque do positive, negative or zero work?
Positive when it acts in the sense of rotation, negative when against it, and zero if the body does not turn (Δθ = 0) or τ = 0.
What is the rotational work–energy theorem?
W_net = ∫τ_net dθ = ½Iω² − ½Iω₀² for a rigid body about a fixed axis. It comes from τ_net = Iα with α = ω dω/dθ.
How is dW = τ dθ derived?
When the body turns through dθ, the point where the force acts moves ds = r dθ along the tangent. Only the tangential component F sin φ does work, so dW = rF sin φ dθ = τ dθ.
How much work does a friction torque of 0.30 N·m do over 10 revolutions?
Convert to radians: 10 rev = 20π rad. W = −0.30 × 20π ≈ −19 J, negative because friction opposes the rotation.
Do equal works on two wheels give equal angular speeds?
No. Equal work gives equal ΔK. The final ω also depends on each wheel's rotational inertia.
Why are torque and work both in N·m but not interchangeable?
Torque is a vector measured in N·m. Work is a scalar energy and should be written in joules. Do not give torque in J.

6.3 Angular Momentum and Angular Impulse

What is the angular momentum of a rigid body?
L = Iω about a fixed axis, with I about that axis. L takes the sign of ω. Unit kg·m²/s (equal to N·m·s).
What is the angular momentum of a point object?
L = r × p about a chosen point O. Its size is mvd, where d is the perpendicular distance from O to the line of motion. In the plane, L_z = xp_y − yp_x.
Does an object moving in a straight line have angular momentum?
Yes, about any point not on its path, with size mvd. About a point on its path, L = 0.
What is angular impulse?
∫τ dt, the signed area under the τ–t graph, in the direction of the torque. For a constant torque it is τΔt.
How is net torque related to angular momentum?
τ_net = dL/dt, the slope of the L–t graph. Integrating gives the rotational impulse–momentum theorem: ∫τ_net dt = L₂ − L₁.
What is the difference between the area under τ–θ and the area under τ–t?
Area under τ–θ is work (energy change). Area under τ–t is angular impulse (change in angular momentum).
Does zero torque mean zero angular momentum?
No. Zero net torque means zero slope of L–t, so L is constant, not zero.
A net torque τ = 2.0t (N·m) acts from 0 to 3.0 s. What is ΔL?
ΔL = ∫₀³ 2.0t dt = [t²]₀³ = 9.0 N·m·s.

6.4 Conservation of Angular Momentum

When is the angular momentum of a system conserved?
When the net external torque about the chosen axis is zero: dL_sys/dt = τ_ext. Forces through the axis or parallel to it give no torque about it.
Why do internal torques cancel?
Third-law pairs act along the line joining two particles, so (r₁ − r₂) × F₁₂ = 0. Two interacting parts give each other equal and opposite angular impulses.
What happens to ω when a spinning system changes shape with no external torque?
I₁ω₁ = I₂ω₂, so ω rises as I falls. Constant L does not mean constant ω.
What happens to kinetic energy when a freely spinning system pulls in its mass?
K = L²/(2I) at constant L, so K ∝ 1/I. Halving I doubles K; the extra energy comes from internal work. Kinetic energy is not conserved even though L is.
What is the final ω when two parts on one shaft lock together?
ω_f = (I_Aω_A + I_Bω_B)/(I_A + I_B), with signs for opposite senses. A 0.10 kg·m² disk at 12 rad/s locked to a 0.050 kg·m² disk at rest gives 8.0 rad/s.
Why is linear momentum not conserved when something hits a pivoted bar?
The pivot exerts an external force during the collision. That force passes through the axis, so it has no torque about the pivot and angular momentum about the pivot is conserved instead.
Does friction destroy angular momentum?
No. Friction transfers angular momentum to the surroundings. Total angular momentum is conserved in all interactions.
How do you find the total angular momentum of a system of parts?
L_sys = ΣIᵢωᵢ + Σmᵢvᵢdᵢ, all about one axis, with signs for parts turning the other way.

6.5 Rolling

What is the condition for rolling without slipping?
The arc that touches the ground equals the distance moved: Δx_cm = rΔθ, so v_cm = rω and a_cm = rα. These hold only when not slipping.
What are the speeds of the contact point and the top of a rolling wheel?
The contact point is momentarily at rest. The center moves at v_cm and the top at 2v_cm.
What is the kinetic energy of an object rolling without slipping?
K = ½Mv_cm² + ½I_cm ω² = ½M(1 + β)v_cm², where I_cm = βMr². β is 1 for a hoop, 2/3 for a spherical shell, 1/2 for a solid cylinder and 2/5 for a solid sphere.
What is the acceleration of an object rolling down a slope without slipping?
a_cm = g sin θ / (1 + β), where I_cm = βMr². Objects with smaller β (more mass near the axis) accelerate faster.
How fast is an object moving after rolling from rest through a height drop h?
v = √(2gh / (1 + β)). A solid sphere (β = 2/5) has v = √(10gh/7). Mass and radius cancel.
Does static friction do work on an object rolling without slipping?
No. It acts at the contact point, which is momentarily at rest, so it dissipates no energy. It provides the torque that keeps ω matched to v_cm/r.
How do you analyse a rolling object that is slipping?
v_cm ≠ rω. Use F_net = Ma_cm and τ_net = I_cm α separately, with kinetic friction μ_k N opposing the sliding of the contact point. Slipping stops when v_cm = rω.
What friction is needed to roll without slipping on a slope?
f = βMg sin θ / (1 + β), so rolling needs μ_s ≥ β tan θ / (1 + β). Find f from Newton's laws; do not assume f = μ_s N.

6.6 Motion of Orbiting Satellites

What are the conserved quantities for a satellite orbiting a much larger body?
Mechanical energy E = ½mv² − GMm/r, because gravity is the only force and is conservative; and angular momentum about the central body, because gravity points along r and exerts no torque about the center.
What is the speed of a satellite in a circular orbit?
v = √(GM/r), from GMm/r² = mv²/r. A higher orbit is slower.
What are K, U and E for a circular orbit?
K = GMm/(2r), U = −GMm/r and E = −GMm/(2r). So K = −½U and E = ½U = −K.
How do speed and distance relate at the closest and farthest points of an elliptical orbit?
Angular momentum is conserved and v ⟂ r at those two points, so r_p v_p = r_a v_a. The satellite is fastest at the closest point.
What stays constant in an elliptical orbit?
Only the total energy E and angular momentum L. K and U change: K is largest and U lowest at the closest point. L = mvr only at the two ends.
What is escape speed from a distance r?
v_esc = √(2GM/r) = √2 × the circular orbit speed at r. A body with E = 0 just reaches infinity. The satellite's mass cancels.
How is U_g = −GMm/r derived from the force?
Integrate the force from infinity, where U = 0: U(r) = −∫ F_r dr with F_r = −GMm/r², giving U = −GMm/r, negative for all finite r.
What changes E and L of a satellite?
An outside force such as a thruster firing or drag. With gravity alone, both are constant.

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