 Torque and Angular Acceleration
 When a rigid object has a net torque (≠0), it undergoes an angular acceleration a
 The angular acceleration is directly proportional to the net torque
 The relationship is analogous to ∑F = ma (Newton’s Second Law)
 a = F/m
 α = a/r
 Rotational Inertia, I
 The analogy of the mass in a rotating system: the rotational inertia (moment of inertia) of the point
mass
 SI units: kg m2
 Newton’s Second Law for a Rotating Object
 The angular acceleration is directly proportional to the net torque and inversely proportional to the
moment of inertia (rotational inertia) of the object
 Difference between moment of inertia and mass:
Moment of inertia depends on:
 the quantity of matter (mass)
 the distribution of matter in the rigid object
 the location of the axis of rotation
 Moment of Inertia of a Uniform Ring with mass M
 The hoop is divided into a number of small segments (m1, m2, …)
 These segments are equidistant from the axis
 Other
Moments
of Inertia
 ex1
 Find the moment of inertia of two equal masses at equal distances from the axis of rotation
 ex2
 If the dumbbell-shaped object is rotated about one end, is its moment of inertia
less than, c) the same as the moment of inertia about its center?
a) more than, b)
 ex3
Consider a light rotating rod with 4 equal masses attached to it at distances 2 and 4, as shown. The axis of
rotation is thru the center of the rod, perpendicular to its length. The 4 masses are moved to new positions so
that all masses are now 3 units from the axis.
Did the moment of inertia
2/7/2016
Document1
1
a) increase,
b) decrease,
c) remain the same?
 rolling without slipping
 hollow sphere (red),
cylinder (blue)
solid sphere (orange),
cylindrical ring (green), solid
 The time for each object to reach the finishing line depends on their moment of inertia
 ex4
 If the hanging blocks are released simultaneously from rest, is it observed that a) the block at the left
lands first, b) the block at right lands first, c) both at the same time?
 ex5
 A light rope wrapped around a disk-shaped pulley is pulled tangentially with a force of 0.53 N. Find the
angular acceleration of the pulley, given that its mass is 1.3 kg and its radius is 0.11 m.
 ex6
 A 0.31 kg cart (M) on a horizontal air track is attached to a string, which passes over a disk-shaped
pulley of mass 0.080 kg and radius 0.012 m, and is pulled vertically downward with a constant force of
1.1 N.
Find tension in the string between the pulley
acceleration of the
and the cart and the
cart.
 Rotational Kinetic Energy, Krot
 An object rotating about some axis with an angular speed, ω, has rotational kinetic energy
Krot =½Iω2
Krot depends on the distribution of mass in the rotating object
The father the mass from the axis of rotation, the greater its kinetic energy
 Rolling Motion
 Kinetic energy of rolling motion
 An object rotating about some axis with an angular speed, ω, has rotational kinetic energy
K = ½mv2 + ½Iω2
Krot depends on the distribution of mass in the rotating object
The father the mass from the axis of rotation, the greater its kinetic energy
 Total Energy of a System
 Conservation of Mechanical Energy
 This is for conservative forces, no dissipative (non-conservative) forces such as friction can be
present
2/7/2016
Document1
2
 ex7
 A solid sphere and a hollow sphere of the same mass and radius roll without slipping at the same speed.
Is the kinetic energy of the solid sphere a) more than, b) less than, or c) the same as the kinetic energy of
the hollow sphere?
 ex8
 A 1.20 kg disk with a radius of 10.0 cm rolls without slipping. If the linear speed of the disk is 1.41 m/s,
find the translational KE, rotational KE and the total KE.
 ex9
 Derive formula for v of a rolling solid sphere at the bottom of the ramp using conservation of
mechanical energy
 Angular Momentum, L
 Object of mass m moving with speed v has linear momentum: p = mv
 Object moving with angular speed ω along a circle with radius r has angular momentum
 L=Iω
SI unit: kg m2/s
 Linear:
 Angular:
 Conservation of Angular Momentum
 The angular momentum of a system is conserved (remains constant) when the net external torque
acting on the systems is zero.
 ex10
 A student sits on a piano stool holding a sizable mass in each hand. Initially, he holds his arms
outstretched and spins about the axis of the stool with angular speed of 3.74 rad/s, his moment of
inertia 5.33 kg m2. He then pulls his arms in to his chest, new moment of inertia 1.60 kg m2. Find his new
angular speed.
 ex11
 A skater pulls in her arms, decreasing her rotational inertia by a factor of two, and doubling her angular
speed. Is her final kinetic energy a) equal to, b) greater than, or c) less than her initial kinetic energy?
 Problem Solving Hints
The same basic techniques that were used in linear motion can be applied to rotational motion.
F becomes τ,
m becomes I
a becomes α,
v becomes ω
x becomes θ
2/7/2016
Document1
3