Unit 5 Beta

Torque and Rotational Dynamics: the one-page sheet

5.1 Rotational Kinematics

Rotation about a fixed axis is described by the angle θ in radians, the angular velocity ω = Δθ/Δt and the angular acceleration α = Δω/Δt, each with a sign for clockwise or counterclockwise. Every point of a rigid system shares them, and for constant α the three linear kinematic equations hold with θ, ω and α in place of x, v and a.

Measure angles in radians: θ = (distance along the circle) ÷ r. One revolution = 2π rad = 360°.
Pick a positive direction, usually counterclockwise. Then θ, ω and α carry signs exactly as x, v and a do on a line.
A rigid system turns as one piece: every point shares the same θ, ω and α.
Constant α: ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2αΔθ.
On an ω-t graph the slope is α and the area is Δθ.

one angle θ describes the whole wheel the wheel has an angular velocity ω = Δθ/Δt, in rad/s the wheel has an angular acceleration α = Δω/Δt, in rad/s² the three kinematic equations hold with θ, ω, α in place of x, v, a the wheel speeds up (or slows down)

rigid system: An object or system that keeps its shape as it moves, so every part of it turns through the same angle in the same time.
axis of rotation: The line about which an object turns. Every point of the object moves in a circle centered on this line.
radian: The SI unit of angle: the angle for which the distance along a circle equals the radius. One revolution is 2π rad, about 6.28 rad, or 360°.
angular displacement: The change in an object’s angle, Δθ = θ − θ₀, in radians, with a sign for the direction of turning (counterclockwise positive is a common choice).
angular velocity: The rate of change of angle, ω = Δθ/Δt, in rad/s. Its sign says whether the object turns clockwise or counterclockwise.
angular acceleration: The rate of change of angular velocity, α = Δω/Δt, in rad/s². When α and ω have the same sign the object speeds up; opposite signs, it slows.
rotational kinematic equations: For constant angular acceleration: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², and ω² = ω₀² + 2α(θ − θ₀). They mirror the linear equations with θ, ω, α for x, v, a.

5.2 Linking Linear and Rotational Motion

A point a distance r from the axis of a turning rigid object travels s = rθ, moves at v = rω and has tangential acceleration a_T = rα, with angles in radians. All points share θ, ω and α, but points farther out move farther and faster. A string or belt that does not slip moves with the rim it touches.

Every point of a rigid object shares θ, ω and α; how far and how fast a point moves depends on r.
s = rθ, v = rω, a_T = rα, with angles in radians.
A point twice as far from the axis travels twice as far and twice as fast.
A string unwinding without slipping moves with the rim: its length, speed and acceleration are rθ, rω, rα.
Centripetal acceleration v²/r = rω² is a different quantity from the tangential rα.

it moves on a circle of radius r as the object turns the point travels a distance s = rθ along its circle the point moves at speed v = rω, tangent to its circle the point’s speed changes at a_T = rα it moves with the rim: same distance, speed and acceleration

arc length: The distance traveled along a circle of radius r while turning through an angle θ in radians: s = rθ.
tangential speed: The speed of a point on a turning object, along its circle: v = rω, with ω in rad/s. Points farther from the axis move faster.

5.3 Torque

Torque measures how well a force turns an object about an axis: τ = r⊥F = rF sin θ, where the lever arm r⊥ is the perpendicular distance from the axis to the force’s line of action. Torques carry a sign for clockwise or counterclockwise and add to a net torque. A force diagram draws each force where it acts.

Torque is the turning effect of a force: τ = r⊥F = rF sin θ, in N·m.
The lever arm r⊥ is the perpendicular distance from the axis to the line of action, not the distance to the point where the force acts.
Only the part of a force perpendicular to r turns the object.
Give each torque a sign: counterclockwise + and clockwise − (or the reverse, stated). Add them for the net torque.
A force diagram starts each force at its point of application, labeled with type and agent.

its line of action may pass the axis at some distance the force has a lever arm r⊥ = r sin θ it exerts a torque τ = r⊥F that tends to turn the object one way their torques add with signs (counterclockwise +, clockwise −) to a net torque its lever arm and its torque are zero

line of action: The straight line through a force’s point of application, along the force’s direction, extended both ways.
torque: The turning effect of a force about an axis: τ = r⊥F = rF sin θ, in N·m. Sign gives the direction, clockwise or counterclockwise. The net torque is the sum with signs.
lever arm: The perpendicular distance r⊥ from the axis of rotation to a force’s line of action: r⊥ = r sin θ. A force whose line passes through the axis has zero lever arm.
force diagram: A diagram of an extended object that draws each force from the point where it acts, labeled with its type and agent, so torques about an axis can be read from it.

5.4 Rotational Inertia

Rotational inertia I measures how hard an object is to set turning about an axis. For point masses I = Σmr², so it depends on both the mass and how far the mass is from the axis, with distance counted squared. The inertias of standard shapes are given. About an axis a distance d from the center of mass, I = I_cm + Md².

Rotational inertia I = Σmr² is the rotational partner of mass: it measures resistance to a change in rotation.
r is each mass’s distance from the axis, and it is squared: doubling r gives 4 times the I.
Same mass, different distribution, different I. A hoop (MR²) beats a disk (½MR²) of the same M and R.
Inertias of disks, rods, hoops and spheres are given on the exam; you do not memorize them.
Parallel-axis theorem: I = I_cm + Md². I is least about the center of mass.

it resists changes in that motion in proportion to its mass it counts more, as the square of its distance gives the rotational inertia I = Σmr², in kg·m² I increases (with r²) I = I_cm + Md², always larger than I_cm

rotational inertia: How hard it is to change an object’s rotation about an axis: I = Σmr² for point masses, in kg·m². It depends on the mass and how far that mass is from the axis. Also called moment of inertia.
mass distribution: How an object’s mass is spread out relative to an axis. The same mass placed farther from the axis gives a larger rotational inertia.
parallel-axis theorem: I = I_cm + Md²: the rotational inertia about any axis equals the inertia about a parallel axis through the center of mass plus the total mass times the square of the distance between the axes.

5.5 Rotational Equilibrium

An object is in rotational equilibrium when the net torque on it is zero, so its angular velocity stays constant; it is in translational equilibrium when the net force is zero. A static object needs both. Take torques about the point where an unknown force (such as a hinge force) acts, and remember that a hinge force can point in any direction.

Rotational equilibrium: Στ = 0, so ω stays constant. Translational equilibrium: ΣF = 0. They are independent; static objects need both.
In equilibrium you may take torques about any point. Choose the point where an unknown force acts (a hinge, a support) to remove it.
A uniform object’s weight acts at its center; it has a torque unless that point is on the axis.
A hinge force usually has both a horizontal and a vertical part. Do not assume its direction.
Draw a force diagram first, each force at its point of application.

its angular velocity does not change (rotational first law) its center of mass does not accelerate it stays at rest: static equilibrium that force drops out of the torque equation the net torque is no longer zero and the object starts to turn

rotational equilibrium: The state of an object whose net torque is zero, so its angular velocity does not change. With zero net force as well, an object at rest stays at rest (static equilibrium).
Newton's first law in rotational form: With zero net torque on it, an object’s angular velocity stays constant: at rest stays at rest, turning keeps turning at the same rate.
hinge force: The contact force a hinge or pivot exerts on an object. Its direction is not fixed in advance; it is whatever balances the other forces, often with both horizontal and vertical parts.

5.6 Newton's Second Law for Rotation

Newton’s second law in rotational form says α = τ_net/I: the net torque sets the angular acceleration, and a larger rotational inertia means a smaller α. Systems with a pulley that has mass need both the translational law for each block and the rotational law for the pulley, linked by a = Rα; the string’s tension is then less than a falling block’s weight.

α = τ_net/I, in the direction of the net torque: the rotational partner of a = F_net/m.
For a system with parts that move and parts that turn, write ΣF = ma for each moving part and Στ = Iα for each turning part, then link them with a = Rα.
A pulley with mass needs a net torque, so the tensions on its two sides differ.
Same torque, larger I (mass farther out): smaller α.
α against τ is a line through the origin with slope 1/I.

its angular velocity changes the angular acceleration is larger: α = τ_net/I the same torque gives a smaller angular acceleration the tension is less than the block’s weight, and the two sides of a string over the pulley carry different tensions a = Rα links the block’s motion to the pulley’s

Newton's second law in rotational form: α = τ_net/I: an object’s angular acceleration equals the net torque on it divided by its rotational inertia, and turns the way the net torque does.
pulley with mass: A pulley whose rotational inertia matters. The string’s tensions on its two sides differ, and their difference times the radius gives the torque that spins it up: (F_T1 − F_T2)R = Iα.