Q1: How large are these
angles in radians? A half rotation, a full rotation, 5 rotations,
45 degrees, 90 degrees, 360degrees.
Half rotation (180°): 180° x 2p
rad / 360° = p rad
Full rotation (360°): 360° x 2p
rad / 360° = 2p rad
5 rotations (5x360°): 1800° x 2p
rad / 360° = 10p rad
45 degrees: 45° x 2p rad / 360°
= p rad / 4
90 degrees: 90° x 2p rad / 360°
= p rad / 2
360 degrees: 360° x 2p rad /
360° = 2p rad
car engine rotates at a rate of 4500 rpm (revolutions per minute).
What is the angular velocity of the engine in radians / sec?
man of height 180 cm subtends an angle of .0036 radians. How far
away is the man?
To solve this problem we must employ the
equation: q = s / r
r = s / q = 1.8m
/ 0.0036 rad = 500 m
The angular position of a merry-go-round
after a switch is thrown is specified as:
are the inital angular position, angular velocity and angular acceleration
of the merry-go-round before the switch is thrown? (Hint: compare
to the linear kinematic equations.)
are the angular position, velocity and acceleration of the merry-go-round
at t = 5 seconds.
a) Angular position: q
(5s) = [1.23 rad + (0.75 rad / s)(5s) + 1/2(0.10 rad / s2)(5s)2]
(5s) = 1.23 rad + 3.75 rad + 1.25 rad = 6.23 rad
b) Angular velocity: w
= wo + a t, which is
related to the linear equation v = vo + at
w = 0.75
rad / s + (0.10 rad / s2
x 5s) = 1.25 rad / s
c) Angular acceleration: remains constant
at 0.10 rad / s2
An elecrical motor running at 1000 rpm is turned
off, and its angular velocity decreases uniformly to 500 rpm over
What is the angular acceleration of the motor?
To solve this problem, we can rearrange the equation w
= wo + at to solve
for a. However, we must first convert
the angular velocities into rad/s as follows:
Now, rearranging w = wo
+ a t to solve for a,
we get: a = (w -
wo) / t
a = (16.7p
rad / s - 33.3p rad / s) / 3s
= -5.5p rad / s2 (The
negative sign indicates that the motor is slowing down.
the motor continues to decelerate uniformly from 500 to 0 rpm,
how many revolutions does the motor turn through before coming
to a rest?
To solve this problem, we must employ
the equation q(t) = qo + wt
+ 1/2a t2 as follows:
= 0 rad + (16.7 rad / s)(3s) + 1/2(-5.5p rad
q(3s) = 50.1 rad - 24.75p rad
= 25.35 p rad = 12.7 revolutions
the motor continues to uniformly coast to a halt, how much time
is required to come to a complete stop?
This question can be answered intuitively. If we know that
the motor is decelerating uniformly, and it took 3 seconds to
go from 1000 rpm to 500 rpm, then we know that it will take an
additional 3 seconds to reach a halt.