1

Laws of Angular Motion

Objectives:

• Define internal & external torques, rigid bodies

• Understand and apply the 3 laws of angular motion

• Define angular impulse and understand the relationship between angular impulse and angular momentum

• Introduce the concepts of angular work, power, & rotational kinetic energy

Internal vs. External Torque• Internal Torque : is applied to a system by a force

acting within the system

• External Torque : is applied to a system by a force or torque acting across the boundary of the system

Fgastroc

Fquads

System

Wleg Wfoot

Tflexor

Rigid Body• An object whose change in shape is negligible.

• Objects made up of multiple parts can be considered a rigid body if the parts don’t move relative to one another.

• Example: the leg + foot is a rigid body if no motion (or very little motion) occurs at the ankle

• The laws of angular kinetics that follow apply only to rigid bodies

• In non-rigid bodies, each rigid part making up the body must be analyzed separately

1st Law (Law of Inertia)

• A rigid body in rotation will maintain a constant angular velocity unless acted upon by an external torque.

• If there is no net external torque acting on a rigid body:

– if the body is not rotating, it will continue not to rotate.

– if the body is rotating, it will continue to rotate at a constant velocity (i.e. at the same speed in the same direction)

2

2nd Law (Law of Acceleration)

where:– T : net external torque about the COM (or axis)– I : body’s moment of inertia about the COM (or axis)– α : angular accel. of the body about the COM (or axis)

• If there is a net external torque acting on a body, the angular acceleration is:– directly proportional to the net torque– inversely proportional to the moment of inertia– in the direction of the net torque

T = I α

• For rotation of a rigid body about its center of mass (or a fixed axis):

Effects of Torque

Decrease in – dir.(+)(–)

Increase in – dir.(–)(–)

Decrease in + dir.(–)(+)

Increase in + dir.(+)(+)

Change in VelocityTorqueVelocity

• Net torque and angular velocity ω in same direction:magnitude of angular velocity increases

• Net torque and angular velocity ω in opposite direction:

magnitude of angular velocity decreases (deceleration)

3rd Law (Law of Reaction)• For every action, there is an equal and opposite

reaction.

• If the forces acting across a joint between two bodies causes body 1 to experience a torque, body 2 will experience a torque:

– of the same magnitude

– in the opposite direction

femur

tibia

Mextension

Mextension

Example Problem #1During a squat lift, a person is holding a 450 N weight

as shown below. What resultant hip moment is required for the lifter to remain motionless?

If the hip extensors have an average moment arm of 5 cm, what total force do they need to generate?

W = 450 N40 cm

HIP

What = 430 N

15 cm

3

Example Problem #2During a sit-up, the hip flexors generate a torque of

85 Nm on the head-arms-torso. What torque do they generate on the lower limbs?

Given the body position and inertial properties shown below, what are the accelerations of the head-arms-torso and lower limbs?

W = 220 NW = 465 N

Fgrf

40 cm15 cmIlower = 6.0 kg m2Ihat = 11.0 kg m2

Radial & Tangental Acceleration• The acceleration of a body in angular motion can

be resolved into two components:

at– Tangental: along

path of motion

– Radial: perpendicular to path of motion

ω

ar

α

a

ar =v2

r= r ω2

at = r α

r

v

Torques & Tangental Acceleration

• Torques produce tangental acceleration only

at

α

at = r α

r

T = I αat =

Ir

T

T• Radial acceleration must

come from some other source!

Centripetal Force• Centripetal force produces radial acceleration

• Magnitude of centripetal force:

Fc = m ar =m v2

r= m r ω2

ω

ar

r

v

Fc

m

• Force required increases with:– object mass (m)– velocity (v or ω)– distance (r) from axis of

rotation

• Fc always directed inward towards the axis of rotation axis of rotation

4

Example Problem #3A 3500 lb. race car is attempting to go through a flat

turn of radius 500 ft. at 100 mi/hr.

What total friction force between the road and tires is required?

If the coefficient of friction between the road and tires is 1.0, will the car be able to negotiate the turn?

Angular Impulse• The linear motion of a body depends both on the

force and the duration that the force is applied

• The angular motion of a body depends both on the torque and the duration that the torque is applied

• Angular Impulse : a measure related to the net effect of applying a torque (T) for a time (t):

Angular Impulse = T t

• Angular impulse increases with:– Increased torque magnitude– Increased duration of application

Angular Impulse & Momentum

• The angular impulse due to the net external torque acting on a system equals the change in the angular momentum of the system over the same period of time

Ang. Impulse = Texternal (t2 – t1) = I2 ω2 – I1 ω1

ang. impulse when Texternal is constant between t1 and t2

angular momentum at time t1angular momentum at time t2

Example Problem #4A hammer thrower is able to apply an average torque

of 100 Nm to the hammer while spinning about his longitudinal axis.

The ball of the hammer has a mass of 7.25 kg and spins at a distance of 1.5 m from the axis of rotation

If the hammer ball starts from rest, what is its angular velocity after 3 s, just prior to release?

What is the magnitude of its linear velocity upon release?

5

Work, Power, & Energy

Kinetic Energy

Power

Work

KEa = ½ I ω2KE = ½ m v2

Pa = Wa / ∆tP = W / ∆t

Wa = T θW = Fx∆px + Fy∆py

AngularLinear

• The concepts of work, power and kinetic energy also apply to objects in rotation:

• General relationship between work and energy:

W + Wa = ∆KE + ∆KEa + ∆PE + ∆TE