Unit 5 Foundations: How Forces Create Rotation

Torque (τ)

The rotational effectiveness of a force about a pivot/axis; depends on both the force and where/how it is applied.

Pivot (axis of rotation)

The point or line about which an object rotates (e.g., hinge, axle); torques are computed about this axis.

Radius / position vector (r) in torque

The distance from the pivot to the point where the force is applied (measured from the axis to the point of application).

Angle in torque (θ)

The angle between the radius vector r (from pivot to application point) and the force vector F.

Torque magnitude formula

τ = rF sinθ, where only the component of force perpendicular to r produces torque.

Perpendicular component of force

The part of a force that is perpendicular to the radius vector; equals F sinθ and is the only part that creates torque.

Zero-torque condition (force along radius)

If θ = 0° or 180°, then sinθ = 0 and the force produces no torque (push/pull directly toward/away from the pivot).

Lever arm (moment arm) (r⊥)

The perpendicular distance from the pivot to the force’s line of action; used so τ = F r⊥.

Line of action

The infinite line in the direction of a force passing through its point of application; used to find the lever arm (shortest distance to pivot).

Sign convention for torque

Assign clockwise and counterclockwise torques opposite signs (often CCW positive, CW negative) and apply consistently when summing torques.

Net torque (τnet)

The signed sum of all external torques about a chosen axis; determines angular acceleration.

Rotational dynamics (Newton’s 2nd law for rotation)

τnet = Iα, relating net torque to rotational inertia I and angular acceleration α.

Torque vs energy (common trap)

Torque is not energy: although both can be in N·m, torque is a (pseudo)vector tendency to rotate, while energy is a scalar measured in joules.

Static equilibrium

A state with no linear acceleration and no angular acceleration; requires ΣFx = 0, ΣFy = 0, and Στ = 0.

Torque equilibrium condition

Στ = 0 about a chosen axis; ensures no angular acceleration (no tendency to start rotating) about that axis.

Force equilibrium conditions

ΣFx = 0 and ΣFy = 0; ensure no translational acceleration.

Strategic pivot choice

Choosing the axis through a point where unknown forces act (e.g., hinge) so those forces produce zero torque and drop out of Στ.

Center of mass of a uniform beam

For a uniform beam/rod, weight acts at the midpoint (L/2 from either end), not at the end.

Rotational inertia / moment of inertia (I)

A measure of how hard it is to change an object’s rotational motion about a specific axis; rotational analog of mass.

Axis dependence of rotational inertia

I is not an intrinsic single value for an object; it changes if the axis of rotation changes (center vs end vs shifted axis).

Point-mass definition of rotational inertia

I = Σ mi ri², summing each mass times the square of its perpendicular distance to the axis.

Thin hoop (ring) moment of inertia

About its center axis perpendicular to the plane: I = MR².

Solid disk/solid cylinder moment of inertia

About its central symmetry axis: I = (1/2)MR².

Uniform thin rod moments of inertia

About center, perpendicular to rod: I = (1/12)ML²; about one end, perpendicular to rod: I = (1/3)ML².

Mass distribution effect (r² effect)

Mass farther from the axis increases I dramatically because distance enters as r²; moving mass inward decreases I and can increase rotational speed if external torque is small.