AP Physics 1 Chapter 8 and 9 Notes
A rigid body is an object with a definite shape that does not change. A steel ball is a rigid body, pudding is not.
The motion of a rigid body can be analyzed as the translational motion of the object’s center of mass (momentum) and
the rotational motion about its center of mass.
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The axis of rotation
is perpendicular to the page and passes through “O”.
The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same
time (they sweep out the same area).
The angle θ in radians is defined:
where l is the arc length. Radians are dimensionless. THIS EQUATION ONLY WORKS WHEN THE ANGLE IS IN RADIANS!
360o = 2π radians.
Example 8-1 (pg. 195) A bike wheel rotates 4.50 revolutions. How many radians has it rotated?
Angular displacement:
Symbol ϴ
Unit: rad
Angular Velocity
Symbol: ω
Unit: rad/s
Direction: use RHR
Angular Acceleration
Symbol: α
Unit: rad/s2
Direction:
Every point on a rotating body has an angular velocity ω and a linear velocity v.
Equation:
𝑣=𝑟𝜔
Example 8-3 (pg. 198)
Is the lion faster than the horse?
On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another
child sits on a lion halfway out from the center. (a) Which child has the greater linear velocity? (b) Which
child has the greater angular velocity
If the angular velocity of a rotating object changes, it has a tangential acceleration
Even if the angular velocity is constant, each point on the object has a centripetal acceleration
The frequency is the number of complete revolutions per second (Hz)
The period is the time one revolution takes
Example 8-5 (pg. 200) The platter of the hard drive of a computer rotates at 7200 rpm (rpm =
revolutions per minute = rev/min). (a) What is the angular velocity (rad/s) of the platter? (b) If the
reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point
on the platter just below it?
Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with
the substitution of the angular quantities for the linear ones.
Example 8-6 (pg. 201) A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. (a) What is its
average angular acceleration? (b) Through how many revolutions has the centrifuge rotor turned during
its acceleration period, assuming constant angular acceleration?
Example 8-7 (pg. 202) A bicycle slows down uniformly from to rest over a distance of 115 m. Each wheel
and tire has an overall diameter of 68.0 cm. Determine (a) the angular velocity of the wheels at the
initial instant (b) the total number of revolutions each wheel rotates before coming to rest; (c) the
angular acceleration of the wheel; and (d) the time it took to come to a stop.
# 18 (pg. 219) A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How
far will a point on the edge of the wheel have travelled in this time?
Rotational Dynamics
What causes rotational motion? An applied force causes motion therefore an applied force is necessary
to make a body rotate but where that force is applied will also determine the rotation of the body.
Torque - The quantitative measure of the tendency of a force to cause or change the rotational motion
of a body.
“Angular force”
Unit: m•N
Direction: Use Right Hand Rule
𝜏=𝑟𝐹𝑠𝑖𝑛 𝜃
Example: Rank the situation from highest torque to lowest torque.
Example: Two thin disk-shaped wheels, of radii RA = 30 cm and RB = 50 cm, are attached to each other
on an axle that passes through the center of each, as shown. Calculate the net torque on this compound
wheel due to the two forces shown, each of magnitude 50 N.
Rotational Inertia
Just like mass (inertia) plays a huge role on the resulting acceleration when a net force is applied to an
object, rotational inertia plays a huge role when a torque is applied to an object.
Rotational inertia (sometimes called the moment of inertia) is the tendency of a body to resist a
rotation.
I  m1r12  m2 r22  ...   mi ri 2
i
I (rotational inertia) measured in kg·m2
The rotational inertia of an object depends not only on its mass distribution but also the location of the
axis of rotation—compare (f) and (g), for example.
Example 8-10 (pg. 207): Two small “weights,” of mass 5.0 kg and 7.0 kg are mounted 4.0 m apart on a
light rod (whose mass can be ignored), as shown in the figure. Calculate the moment of inertia
(rotational inertia) of the system
(a) when rotated about an axis halfway between the weights
(b) when rotated about an axis 0.5 m to the left of the 5.0-kg mass.
Example 8-11 (pg. 209) A 15.0 N force is applied to a cord wrapped around a pulley of radius 33.0 cm.
The pulley reaches an angular speed (w) of 30.0 rad/s in 3.00 s. Since this is a real pulley, there is a
frictional torque (tfr= 1.10 m-N) opposing rotation.
A) Calculate the net torque.
B) Calculate the angular acceleration
C) Calculate the moment of inertia of the pulley.
Rotational Kinetic Energy
The kinetic energy of a rotating object is given by
By substituting the rotational quantities, we find that the rotational kinetic energy can be written:
A object that both translational and rotational motion also has both translational and rotational kinetic
energy:
When using conservation of energy, both rotational and translational kinetic energy must be taken into
account.
All these objects have the same potential energy at the top, but the time it takes them to get down the
incline depends on how much rotational inertia they have.
Example 8-13 (pg. 211) If the solid sphere rolls without slipping, determine the speed at the bottom of
the incline.
Angular Momentum
In analogy with linear momentum, we can define angular momentum L:
Therefore, systems that can change their rotational inertia through internal forces will also change their
rate of rotation
Example 8-15 (pg. 214) A small mass m attached to the end of a string revolves in a circle on a
frictionless tabletop. The other end of the string passes through a hole in the table. Initially, the mass
revolves with a speed v1 = 2.4 m/s in a circle of radius R1 = 0.80 m. The string is then pulled slowly
through the hole so that the radius is reduced to R2 = 0.48 m. What is the speed, v2, of the mass now?