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Ch. 8 Rotational Kinematics
8.1 Rotational Motion
Ch. 2 focused on Kinematics in 1-D, describing the motion of an object as it
moves linearly – along a straight line.
In Ch. 5 we studied uniform circular motion, where an object moves along a
circular path at constant speed.
Now let’s analyze objects moving on a circular path, where the speed can be
changing, thus the object can be accelerating.
What is a circle???
A circle is a locus of points in a plane equidistant from a fixed point in the plane.
In rotational motion, the center of the circular motion becomes the axis of
rotation.
In Ch. 2 we defined the change in linear displacement as:
We can define the change in the angular displacement as:
*This is the angle swept out by a rigid body rotating in a plane about a fixed axis
of rotation which is perpendicular to that plane.
The “vector nature” of the angular displacement is
associated with the direction of rotation:
Δθ is positive for counterclockwise (ccw) rotations.
Δθ is negative for clockwise (cw) rotations.
Units? Radians [rad] 1 revolution = 2 π rad
We could also use degrees: 1 revolution = 360
o
2 π rad = 360
o
As an object rotates, it traces out an arc of length s , where:
Rearrange the equation for arc length:
Notice, θ must be unitless! Radians have no
effect on other units.
Look at two circles with different radii from above:
r 1
r 2
s 2
s 1
In both case the angular displacement Δθ = π/2, but
the arc lengths are different, since the radii are
different.
8.2 Angular Velocity and Acceleration
In Ch. 2 we defined velocity as
We can do the same thing for rotational motion:
Units?
[rad/s]
Units could also be deg./sec., rev/min, etc.
The angular velocity, ω, is positive for ccw rotations and negative for cw
rotations.
And like we did for linear motion, we can also define the instantaneous angular
velocity:
Angular Velocity
Angular Acceleration
What if the angular velocity, ω, is not constant?
In linear motion, a change in velocity → acceleration.
The same holds true for rotational motion
Units?
[rad/s
2
]
Angular Acceleration
The angular acceleration, α, is positive for ccw rotations and negative for cw
rotations.
And like we did for linear motion, we can also define the instantaneous angular
acceleration:
Linear and Rotational Kinematics
There is a beautiful one-to-one correspondence between the equations for linear
motion and the equations for rotational motion. All you do is replace x with θ, v
with ω, and a with α.
Example: The drawing shows a device that can be used to measure the speed of a
bullet. The device consists of two rotating discs separated by a distance d = 0.850 m,
and rotating with an angular speed of 95.0 rad/s. The bullet first passes thru the left disk
and then thru the right disk. It is found that the angular displacement between the two
bullet holes is θ = 0.240 rad. What is the speed of the bullet?
θ
Projection of bullet
hole thru first disk.
The speed of the bullet is given by:
Thus, we need to find the time.
We are given ω and Δθ. From this we can find the time:
Thus,
8.4 The Relationship Between Angular and Tangential Variables
We know that
If we hold r constant and divide by Δ t , we get:
This is the relationship between the tangential velocity and the angular velocity.
P
Q
r P
r Q
v TQ
v TP
As the disk rotates, ω is the same for both P and Q, i.e. they
each sweep out the same angle in the same time.
But, the tangential speeds v
TP
and v
TQ
are different, since the
radii of each point is different.
Notice that v
TQ
is larger, since r
Q
r
P
Take our equation above for tangential velocity and divide by
Δ t again:
This is the relationship between the tangential acceleration and the angular
acceleration.
Rolling without slipping
As you drive in your car at constant speed, your car moves forward, and at the
same time, your tires rotate.
To an observer on the ground, the velocity of a point on the tire is twice the car’s
speed!
Also, whatever distance the car
travels must be equal to the arc
length rotated thru along the edge of
the tire.
Ch. 9 Rotational Dynamics
9.1 Torque
Forces cause linear accelerations:
But what causes angular accelerations, α?
In other words, what is the rotational analog of force?
It’s Torque!
x
F
r
θ
Axis of rotation
r is the distance from the axis of rotation to the point of
contact of the force.
θ is the angle between r and F.
Torque is positive for ccw rotations and negative for cw
rotations.
Units?
[Force x distance]
[N · m]
James finds it difficult to muster enough torque to turn the stubborn bolt
with the wrench. He has a section of rope which he ties to the wrench
as shown and pulls just as hard. Will the torque be increased?
Clicker Question
1. Yes
2. No
Example: You are installing a new spark plug in your car, and the manual specifies
that it be tightened to a torque that has a magnitude of 32 N m. Using the data in the
drawing, determine the magnitude F of the force that you must exert on the wrench.
Example: What is the net torque produced by the forces F
1
and F
2
bout the rotational
axis shown in the drawing? The forces are acting on a rigid rod, and the axis of rotation
is perpendicular to the page. Include both magnitude and direction.
x
F
2
= 40.0 N
F 1
= 10.0 N
Axis
1.15 m 2.70 m
27
o
From the figure, r
1
= 1.15 m, and r
2
= 2.70 m.
1
o
θ 1
= 90
o
2
o
θ 2
= 117
o
F
1
produces a negative (cw) torque ( τ
1
F
2
produces a positive (ccw) torque ( τ
2
Since the net torque is positive, the rod rotates ccw.
