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KINE 3301
Biomechanics of Human Movement
Angular Impulse
Chapter 13
𝜏 𝑡 = 𝐼𝜔𝑓 − 𝐼𝜔𝑖
𝜏 𝑑𝑡 = 𝐼𝜔𝑓 − 𝐼𝜔𝑖
Linear and Angular Momentum
• Recall from chapter 8 that linear momentum was defined as
mass x velocity.
𝑝 = 𝑚𝑣
• Angular momentum is defined as moment of inertia x angular
velocity
𝐿 = 𝐼𝜔
• Angular momentum is a vector , the direction of the vector is
defined by the right hand rule.
• The units for angular momentum are kg∙m2/s
• Angular momentum describes the quantity of rotation that a
body possesses.
Computing Angular Momentum
The diver shown below is rotating with an angular
velocity (𝜔) of −4.2 r/s and a moment of inertia (I) of 5
kg∙m2, compute the magnitude of the divers angular
momentum.
Angular Impulse – Angular Momentum Relationship
• The angular impulse – angular momentum relationship is
derived from Newton’s law of angular acceleration:
Στ = Ι𝛼
𝜔𝑓− 𝜔𝑖
Στ = Ι
𝑡
𝜔𝑓− 𝜔𝑖
𝛼=
𝑡
Στ(t) = I𝜔𝑓 − 𝐼𝜔𝑖
Angular Impulse = change in angular momentum
If you apply a torque over time (angular impulse), it changes the
angular momentum of an object. An angular impulse is
required to make an object spin or stop an object from spinning
An average torque of 3.72 N∙m is applied for a time of t = .9
s to the rigid bar shown below. Determine the final angular
velocity of the bar.
A constant torque of 4.55
N∙m is applied to a rigid bar
with a moment of inertia of
0.8 kg∙m2 and an initial
angular velocity of 0.0 r/s,
determine the final angular
velocity.
What was the angular
impulse that was applied?
To compute the angular impulse
for a non-constant torque the
torque must be integrated with
respect to time:
𝜏 𝑑𝑡 = 𝐼𝜔𝑓 − 𝐼𝜔𝑖
After the torque is applied to the
rigid bar the angular velocity
increases to 5.27 r/s. If the rigid bar
has a moment of inertia of 0.8
kg∙m2 what was the final angular
momentum?
If the torque was applied for t = 0.9
s, what was the angular impulse?
Principle of
Conservation
of Angular
Momentum
When gravity
is the only
force acting on
a system the
angular
momentum is
constant.
During the force application
phase the gymnast applies
forces against the ground.
The reaction forces cause
torque about the gymnast’s
center of mass. The net
positive torque creates
positive angular momentum.
When the gymnast is in the air the
net torque about the center of mass
is zero and as a result angular
momentum is constant.
Is the moment of inertia constant?
Is the angular velocity constant?
Use Excel to compute the angular impulse for each positive and
negative torque phase for the torque curve shown below.
A Tale of Two Divers
Shaquille O’Neal
Height 2.16 m
Mass 147.42 kg
ICM = 29.58 kg∙m2
Guo Jingjing
Height 1.63
Mass 49 kg
ICM = 5.6 kg∙m2
Each diver will perform a layout back 1 ½ rotation dive from a 10 m
platform, they will be in the air for t = 1.68 s. Compute the
following: 1.) angular velocity required to complete 1 ½ rotations
(9.42 rad), 2.) angular momentum required, 3.) angular impulse
required.