Torque and Rotational Energy

torque due to a force

if a force is applied perpendicular to the object wished to be rotated from rest, the torque τ is a vector quantity whose magnitude is given by: τ = Fp r

• r is the distance from the axis of rotation

• torque is proportional to r and the perpendicular force applied to rotate the object

• the units of torque are in Newton-meters, not Joules

if a force is applied at any angle θ relative to the object wished to be rotated from rest, the torque τ is a

vector quantity whose magnitude is given by: τ = F r sin θ

• Fsinθ is the perpendicular component of the applied force which does the job of rotating

• Fcosθ is the parallel component of the applied force which pushes against the hinge

torque is thus defined as the

perpendicular component of force (Fsinθ) times the actual distance from the axis of rotation (r)

• if θ is smaller, the perpendicular force Fsinθ will also be smaller, until θ = 0, thus torque will also be 0

• torque will always be perpendicular in direction to the plane defined by radius and force, since the vector quantity τ = the vector r x the vector F

torque also equals the

force applied (F) x the lever arm (l) which measures effective distance

• the greater the lever arm, the greater the torque

• τ = F l = Fsinθ r, since l = rsinθ

• increasing θ increases the lever arm, and when θ = 90°, the lever arm is at its maximum and equal to r

direction of torque

• if τ > 0, there is counterclockwise rotation from rest since the angles are increasing

• if τ < 0, there is clockwise rotation from rest since the angles are decreasing

net torque

when multiple torques act about the same axis of an object, the net torque is the vector sum of individual torques: τnet = τ1 + τ2 + τ3 + … + τn

• in the case of a rotation about a fixed axis, the torque is always parallel to the axis of rotation, so it can only be a vector pointing forward or backward along the axis

rotational inertia

an object tends to preserve its state of no rotation or rotation at a constant angular velocity unless compelled to change that state by a net external torque

• consequence of Newton’s 1st law (law of inertia)

• net torque results in angular acceleration, since the force applied results in a tangential acceleration

  • Ft gives at, τ gives α

the moment of inertia for a rigid object about an axis quantifies the

tendency to rotate with constant angular velocity about that axis

• determined by the distribution of mass about an axis of rotation

• it is the rotational analog of inertial mass

for regular shapes, the moment of inertia I is given by

I = bmR²

• b is a dimensionless coefficient between 0 and 1

• m is the total mass of the object

• R is the characteristic dimension of the object

for any shape, the moment of inertia I is

I = Σ mi ri²

• a solid distribution of mass has a lower moment of inertia than a hollow distribution of mass

• solid cylinder: I = ½ mR²

• hollow cylinder: I = mR²

• solid sphere: I = 2/5 mR²

• hollow sphere: I = 2/3 mR²

• rod rotating in the center: I = 1/12 mL²

• rod rotating at the end: I = 1/3 mL²

the net torque τ needed to change the angular velocity of a rigid object is proportional to the

moment of inertia I about the axis of rotation and the desired angular acceleration α

• τnet = Iα

• F = m at, and since at = rα, F = m rα, and F ri = (m rα) ri, thus τi = m r²α

• I = Σ mi ri², thus Σ τi = (Σ mi ri²) α, thus τnet = Iα

if the bar is rigid, the

same force will be applied across the bar

• torque will be different because the distance from the axis of rotation (r) is not the same at different points

increasing the length of the bar gives

more lever arm to the force, thus increasing the torque, and this will also increase the moment of inertia since the moment of inertia I is proportional to L²

• increasing the length of the plank will make it accelerate less

the kinetic energy of an object performing a rigid rotation about a fixed axis is

K = Krot = ½ I w²

• if the axis is fixed, any point on the axis will have a zero kinetic energy, since it will not be moving at all, since the angular velocity w = 0

the kinetic energy of a rigid object is due to the

translation of its center and rotation about an axis

• ex. a rolling object, which performs both a translation and a rigid rotation about a moving axis has a kinetic energy K = Ktra + Krot = ½ mv² + ½ Iw²

  • v is the speed of the center of mass of the object

if angular velocity is changed, then rotational kinetic energy is

also changed, and work is related to the change

work-energy theorem for rotations

the work done by the net external torque acting on an object is equal to the change in its rotational kinetic energy, thus Wτ = ∆Krot = ½ Iwf² - ½ Iw0²

• if an object is rotated at varying angular speed, the torque will do work

  • if the angular speed increases, Wτ > 0 and the torque will do positive work

  • if the angular speed decreases, Wτ < 0 and the torque will do negative work

if torque remains constant, the work done by torque is

Wτ = τ ∆θ

• the work done by torque is equal to the magnitude of the torque times the change in angular position or angular displacement from rotation