The document discusses internal and external torques, the three laws of angular motion, and key concepts related to rotational dynamics including: 1) Internal torque is applied within a system while external torque is applied across the system boundary. 2) The three laws of angular motion describe how torques cause changes in angular velocity and acceleration according to an object's moment of inertia. 3) Key concepts like angular impulse, work, power, and kinetic energy can be analyzed similarly to linear motion but involve torque, angular velocity/acceleration, and moment of inertia rather than force and linear velocity/acceleration.

1. Internal vs. External Torque
• Internal Torque : is applied to a system by a force
Laws of Angular Motion acting within the system
• External Torque : is applied to a system by a force
or torque acting across the boundary of the system
Objectives:
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• Define internal & external torques, rigid bodies
• Understand and apply the 3 laws of angular motion Tflexor
• Define angular impulse and understand the System
relationship between angular impulse and angular Fgastroc
momentum
• Introduce the concepts of angular work, power, &
rotational kinetic energy
Wleg Wfoot
Rigid Body 1st Law (Law of Inertia)
• An object whose change in shape is negligible.
• Objects made up of multiple parts can be • A rigid body in rotation will maintain a constant
considered a rigid body if the parts don’t move angular velocity unless acted upon by an external
relative to one another. torque.
• Example: the leg + foot is a rigid body if no motion • If there is no net external torque acting on a rigid
(or very little motion) occurs at the ankle body:
– if the body is not rotating, it will continue not to
• The laws of angular kinetics that follow apply only rotate.
to rigid bodies – if the body is rotating, it will continue to rotate
• In non-rigid bodies, each rigid part making up the at a constant velocity
body must be analyzed separately (i.e. at the same speed in the same direction)
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2. 2nd Law (Law of Acceleration) Effects of Torque
• For rotation of a rigid body about its center of mass • Net torque and angular velocity ω in same direction:
(or a fixed axis): magnitude of angular velocity increases
T= Iα • Net torque and angular velocity ω in opposite direction:
where: magnitude of angular velocity decreases (deceleration)
– T : net external torque about the COM (or axis)
Velocity Torque Change in Velocity
– I : body’s moment of inertia about the COM (or axis)
– α : angular accel. of the body about the COM (or axis) (+) (+) Increase in + dir.
• If there is a net external torque acting on a body, the (+) (–) Decrease in + dir.
angular acceleration is:
– directly proportional to the net torque (–) (–) Increase in – dir.
– inversely proportional to the moment of inertia
– in the direction of the net torque (–) (+) Decrease in – dir.
3rd Law (Law of Reaction) Example Problem #1
During a squat lift, a person is holding a 450 N weight
• For every action, there is an equal and opposite
as shown below. What resultant hip moment is
reaction.
required for the lifter to remain motionless?
• If the forces acting across a joint between two If the hip extensors have an average moment arm of 5
bodies causes body 1 to experience a torque, body cm, what total force do they need to generate?
2 will experience a torque:
What = 430 N
– of the same magnitude femur
15 cm
– in the opposite
direction HIP
Mextension
Mextension
W = 450 N
40 cm
tibia
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3. Example Problem #2 Radial & Tangental Acceleration
During a sit-up, the hip flexors generate a torque of • The acceleration of a body in angular motion can
85 Nm on the head-arms-torso. What torque do be resolved into two components:
they generate on the lower limbs?
– Tangental: along at
Given the body position and inertial properties shown path of motion
below, what are the accelerations of the head- v
– Radial: perpendicular
arms-torso and lower limbs?
to path of motion
a
15 cm 40 cm
Ihat = 11.0 kg m2 Ilower = 6.0 kg m2 at = r α ar
v2 α r
ar = = r ω2 ω
r
W = 465 N W = 220 N
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Torques & Tangental Acceleration Centripetal Force
• Centripetal force produces radial acceleration
• Torques produce tangental acceleration only
• Magnitude of centripetal force:
at = r α at m v2
I F c = m ar = = m r ω2 v
at = T r
r m
T =Iα
• Force required increases with: ar
• Radial acceleration must α – object mass (m)
Fc
– velocity (v or ω)
come from some other T r r
– distance (r) from axis of
source! ω
rotation
• F c always directed inward
towards the axis of rotation axis of rotation
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4. Example Problem #3 Angular Impulse
A 3500 lb. race car is attempting to go through a flat • The linear motion of a body depends both on the
turn of radius 500 ft. at 100 mi/hr. force and the duration that the force is applied
What total friction force between the road and tires is • The angular motion of a body depends both on the
required? torque and the duration that the torque is applied
If the coefficient of friction between the road and tires • Angular Impulse : a measure related to the net
is 1.0, will the car be able to negotiate the turn? effect of applying a torque (T) for a time (t):
Angular Impulse =Tt
• Angular impulse increases with:
– Increased torque magnitude
– Increased duration of application
Angular Impulse & Momentum Example Problem #4
• The angular impulse due to the net external A hammer thrower is able to apply an average torque
torque acting on a system equals the change in of 100 Nm to the hammer while spinning about his
the angular momentum of the system over the longitudinal axis.
same period of time The ball of the hammer has a mass of 7.25 kg and
angular momentum at time t1 spins at a distance of 1.5 m from the axis of
angular momentum at time t2 rotation
If the hammer ball starts from rest, what is its angular
velocity after 3 s, just prior to release?
Ang. Impulse = Texternal (t2 – t1) = I2 ω2 – I1 ω1
What is the magnitude of its linear velocity upon
release?
ang. impulse when Texternal is constant between t1 and t2
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5. Work, Power, & Energy
• The concepts of work, power and kinetic energy
also apply to objects in rotation:
Linear Angular
Work W = Fx∆px + Fy∆py Wa = T θ
Power P = W / ∆t Pa = Wa / ∆t
Kinetic Energy KE = ½ m v2 KEa = ½ I ω2
• General relationship between work and energy:
W + Wa = ∆KE + ∆KEa + ∆PE + ∆TE
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