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Kinematics of Uniform Circular Motion
Introduction
 Uniform Circular
motion- object moving in a circle at a constant speed. Magnitude of velocity
remains constant but the direction is continuous it changing.
Magnitude of v1 = Magnitude of v2
The instantaneous velocity is in the direction of the tangent to
the circular path. As particle moves from A to B during /\t, /\v
is a vector from end of v1 to end of v2. /\v
points towards the center.
Therefore the acceleration is called centripetal( center seeking)
acceleration or radial( along radius, toward the center) acceleration.
The vectors v1, v2, and /\v form a triangle
similar to triangle ABC. The angles, /\ ,
are equal.
If the speed is uniform
2 r
v = ---------- where t is the time for 1 revolution
t
An object moving in a circle of radius r with constant speed has an
acceleration whose direction is toward the Center and whose magnitude is
v2
ac = —-
r
The acceleration vector points toward the center, but the velocity vector
points in the direction of the motion, which is tangential to the circle. thus
the velocity and acceleration vectors are perpendicular to each other at every
point for uniform circular motion.
Consider three cases we have examined
Falling objects- acceleration and velocity in the same direction.
Projectiles- acceleration downward and velocity in various directions.
Uniform circular motion- acceleration due to gravity perpendicular to
velocity.
1 revolution = 2r and Period = time for 1 complete revolution.
Table of Contents
Problems
| 1. |
Allie, 55.0 kg, is tied to a rope and is revolving uniformly in a
horizontal circle of radius 3.50 m. She makes exactly 4.00 revolutions per
second. what is the centripetal acceleration?
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| 2. |
The moon's nearly circular orbit around the Earth has a radius of 384,000
km and a period of 27.3 days. What is the acceleration of the Moon toward the
earth?
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| 3. |
Melissa, 50.0 kg, is tied to a rope and is
revolving uniformly in a horizontal circle of radius 5.00 m. If the
centripetal acceleration is 15.0 m s -2, what is the Period of
this motion? How many revolutions per minute? |
Table of Contents
Answers
| 1. |
Allie, 55.0 kg, is tied to a rope and is revolving uniformly in a
horizontal circle of radius 3.50 m. She makes exactly 4 revolutions per
second. what is the centripetal acceleration?
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If there are 4.00 revolution per second, it takes 0.25 seconds for 1
revolution.
2 r
2 3.50 m
v = --------- = ----------------- = 88.0 m/s
t must be the Period of the object.
t
0.25 s
v2 (88.0
m/s)2
ac = ----- = ----------------- = 2210 m s
-2 This
is an extremely high acceleration and would be
r
3.50
m
hazardous to her health.
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| 2. |
The moon's nearly circular orbit around the Earth has a radius of 384,000
km and a period of 27.3 days. What is the acceleration of the Moon toward the
earth?
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24.0 hours 3600 seconds
27.3 days x ---------------- x -------------------- = 2.36 x 106
seconds
1
day
1 hour
384,000 km = 3.84 x 108 m
2r
2 3.84
x 108 m
v = --------- = ------------------------- = 1020
m/s t must be
the Period of the object.
t
2.36 x 106 s
v2 (1020
m/s)2
ac = ----- = ------------------- = 0.00271 m
s -2
r 3.84
x 108 m
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| 3. |
Melissa, 50.0 kg, is tied to a rope and is
revolving uniformly in a horizontal circle of radius 5.00 m. If the
centripetal acceleration is 15.0 m s -2, what is the Period of
this motion? How many revolutions per minute? |
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v2
ac = ----- => v = (ac
r) 1/2 = (15.0 m s -2 x 5.00 m) 1/2
= 8.66 m/s Solved
for v.
r
2r
2r
2 x 5.00 m
v = --------- => t = ------- =
------------------ = 3.63 s
t
v 8.66 m s -1
1
rev 60 s
--------- x --------- = 16.5 rpm (revolutions per minute)
3.63 s 1 min
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Table of Contents
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