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Flashcards · AP® Physics C: Mechanics · Unit 5

Unit 5: Torque and Rotational Dynamics flashcards

46 cards on topics 5.1 to 5.6. Pick a topic or study the whole unit.

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5.1 Rotational Kinematics

What is the relationship between θ, ω and α?
ω = dθ/dt (rad/s), the slope of θ–t; α = dω/dt = d²θ/dt² (rad/s²), the slope of ω–t. Integrate up: ω(t) = ω₀ + ∫₀ᵗ α dt and θ(t) = θ₀ + ∫₀ᵗ ω dt.
When can you use the constant angular acceleration equations?
Only when α is constant: ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2α(θ − θ₀). If α depends on t, integrate.
Why can a rotating rigid body not be modelled as one object?
Its points move in different directions as it turns. If the rotation does not matter, track the center of mass with the object model instead.
Do all points of a rigid body share the same ω?
Yes. Every point has the same θ, ω and α about a fixed axis. Only linear speeds differ, growing with distance from the axis.
How can you tell whether a rotating object is speeding up or slowing down?
State the positive sense first (for example counterclockwise). Same signs of ω and α: spinning faster. Opposite signs: spinning slower. A negative α alone does not mean slowing.
How do you convert rpm to rad/s?
Multiply by 2π/60. 1 revolution = 2π rad = 360°. Convert before any derivative or integral: 450 rpm ≈ 47 rad/s.
How do you find total angle turned, rather than angular displacement?
∫ω dt gives the signed Δθ. For the total angle turned, split where ω = 0 and add the sizes of each part.
From rest at constant α, how do time and angle scale with final ω?
t = ω/α and Δθ = ω²/(2α). Doubling the final ω doubles the time and quadruples the angle.

5.2 Connecting Linear and Rotational Motion

How are arc length, linear speed and tangential acceleration related to rotation?
s = rθ, v = rω and a_T = rα, with θ in radians, ω in rad/s and α in rad/s². Linear quantities are proportional to r for points of the same rigid body.
Does a point on a body rotating at constant ω accelerate?
Yes. It has centripetal acceleration a_c = v²/r = ω²r toward the axis, even though a_T = 0.
What is the total acceleration of a point on a rotating body?
a_T = rα along the tangent and a_c = ω²r toward the axis are perpendicular, so |a| = √(a_T² + a_c²).
How do you link a string unwinding from a drum to the drum's rotation?
If it does not slip or stretch, v_string = Rω and a_string = Rα, where R is the radius at which the string leaves the rim.
How does centripetal acceleration vary across points on a rigid body at fixed ω?
Use a_c = ω²r, so a_c ∝ r. Using v²/r is misleading because v also changes with r.
Why must angles be in radians in s = rθ?
The radian is defined by s = rθ, so the formulas v = rω and a_T = rα hold only with radian measure. Degrees or rpm give wrong answers.
A rope on a 0.15 m drum has acceleration 0.60 m/s² and does not slip. What is α?
α = a/R = 0.60 ÷ 0.15 = 4.0 rad/s².

5.3 Torque

What is torque?
The turning effect of a force about a chosen axis: τ = r × F, with r from the axis to the point where the force acts. Its size is |τ| = rF sin θ, where θ is the angle between r and F, tail to tail. Unit N·m.
What are the two equivalent ways to calculate torque?
τ = rF⊥, using the force component perpendicular to r (F⊥ = F sin θ), or τ = r⊥F, using the lever arm r⊥ = r sin θ, the perpendicular distance from the axis to the line of action.
What is the lever arm?
The perpendicular distance from the axis to the line of action of the force, not the distance to the point where the force is applied.
How do you find torque from components in the xy-plane?
τ_z = xF_y − yF_x. Positive τ_z is counterclockwise (out of the page), negative is clockwise. r = 0.30î m with F = −20ĵ N gives τ_z = −6.0 N·m.
How do you use the right-hand rule for torque?
Point your fingers along r and curl them toward F through the smaller angle; your thumb points along τ. Thumb out of the page means counterclockwise; into the page means clockwise.
Why does the order matter in τ = r × F?
The cross product is anticommutative: F × r = −(r × F). î × ĵ = k̂ but ĵ × î = −k̂, so reversing the order flips the sign.
Why can torques only be added about one axis?
Torque depends on the chosen axis, so the net torque is the signed sum about a single axis. Forces through that axis (such as hinge forces) give zero torque about it.

5.4 Rotational Inertia

What is rotational inertia?
A measure of a rigid system's resistance to changes in its rotation about a given axis. It depends on mass and how far that mass is from the axis. For point masses I = Σmr²; unit kg·m².
How do you find rotational inertia of a continuous object?
I = ∫r² dm. Choose dm so all of it is the same distance from the axis, write dm = λ dx or σ(2πr dr), integrate, then replace λ or σ by M.
What is the rotational inertia of a uniform rod?
About its center, ML²/12. About one end, ML²/3. About a perpendicular axis at x = a from one end, M(L²/3 − La + a²).
What are the rotational inertias of a hoop and a disk about their central axes?
Thin hoop or cylindrical shell: MR², since all the mass is at R. Uniform disk or solid cylinder: ½MR². A hoop has twice the rotational inertia of a disk of the same M and R.
What is the rotational inertia of an annular ring about its central axis?
½M(R₁² + R₂²), found by integrating rings dm = σ 2πr dr from R₁ to R₂.
What is the parallel axis theorem?
I = I_cm + Md², where d is the distance between the axis and a parallel axis through the center of mass. I is smallest about the center-of-mass axis. Always go through I_cm.
What is the rotational inertia of a uniform disk about a perpendicular axis through its rim?
I = ½MR² + MR² = (3/2)MR², by the parallel axis theorem with d = R.
Derive I for a uniform rod about one end.
λ = M/L and dm = λ dx. I = ∫₀ᴸ x²(M/L) dx = (M/L)(L³/3) = ML²/3.

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

What is rotational equilibrium?
Zero net torque on the system, Στ = 0. Then ω is constant, which may be zero (a resting shelf) or not (a fan at steady speed).
What does Newton's first law say in rotational form?
ω stays constant only if Στ = 0. Unbalanced torques mean ω is changing.
Are translational and rotational equilibrium independent?
Yes. A couple gives ΣF = 0 with Στ ≠ 0. A tumbling object in free fall has Στ = 0 about its center of mass with ΣF ≠ 0. Static equilibrium needs both ΣF = 0 and Στ = 0.
Where does gravity act on an extended object for torque?
At the center of mass: τ_grav = −g∫x dm = −Mg x_cm. For a non-uniform rod, find x_cm = (1/M)∫x λ dx first, rather than using the middle.
How do you choose the point for taking torques?
Take torques about a point where an unknown force acts, so that force drops out of the torque equation. If ΣF = 0, the net torque is the same about every point.
What does "about to tip" mean in an equilibrium problem?
The support force on the far side has just reached zero, so the object balances on the remaining support.
Why must pivot forces still be included in ΣF?
A pivot force has no torque about the pivot, but it is still a force and must balance the others in ΣF_x = 0 and ΣF_y = 0.
What is an extended force diagram?
A diagram showing the object's outline with each force drawn starting at the point where it acts, and the axis marked, so lever arms can be measured.

5.6 Newton’s Second Law in Rotational Form

What is Newton's second law for rotation?
α = Στ/I. The angular acceleration is proportional to the net external torque, in the same sense, and inversely proportional to the rotational inertia about that axis.
How is Στ = Iα derived?
For each piece, F_t = m rα. Multiply by r and sum: Στ = (Σmr²)α = Iα. Internal torques cancel in third-law pairs.
How do you handle an object that both translates and rotates?
Write two separate equations: a_cm = ΣF/M for the center of mass and α = Στ/I for the rotation. Pivot forces appear in the first but give no torque about the pivot.
How do you find ω when the net torque varies with time?
α(t) = Στ(t)/I, so ω(t) = ω₀ + (1/I)∫₀ᵗ Στ dt. Δω is the area under the Στ–t graph divided by I.
What is the tension in a string holding a mass that accelerates downward off a wheel?
T = m(g − a), which is less than mg. Link the mass to the wheel with a = rα if the string does not slip.
When is ω greatest for a varying torque?
When Στ = 0, after which the torque reverses and starts slowing the rotation. The greatest ω is not at the greatest torque.
Does α point in the direction of rotation?
Not necessarily. α follows the net torque. If the torque opposes the rotation, α opposes ω and the body slows.
A wheel with I = 0.15 kg·m² has a net torque of 0.60 N·m. What is α?
α = Στ/I = 0.60 ÷ 0.15 = 4.0 rad/s².

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