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Nonuniform Circular Motion
Introduction
 When you first
start revolving a ball on the end of a string around your head, you must give it
tangential acceleration. this is accomplished by pulling on the string with your
hand displaced from the center of the circle.
The tangential component of the acceleration, aT, is equal to
the rate of change of the magnitude of the velocity of the object.
/\v
v2
aT = -— while aC = —--
/\t
r
These are perpendicular to each other. a = ( aC2 + aT2
) 1/2
Since it is a vector you must give the direction.
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Problems
| 1. |
Doug is in a racing car starting from rest
at the start time. It accelerates to a uniform rate of speed of 43.0 m/s
in 12.0 s. He is driving on a circular track with a radius of 600. m.
Assuming constant tangential acceleration, calculate (a) tangential
acceleration, (b) centripetal acceleration at 40.0 m/s, and (c) Total
acceleration at 40.0 m/s.
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| 2. |
Apollo went from rest to 20.0 m/s in 5.00
seconds on the ice skating short track at the 2002 Olympics.
Assuming constant tangential acceleration, calculate (a) tangential
acceleration, (b) centripetal acceleration at 4.00 seconds, and (c) Total
acceleration at 4.00 seconds. |
Table of Contents
Answers
| 1. |
Doug is in a racing car starting from rest
at the start time. It accelerates to a uniform rate of speed of 43.0 m/s
in 12.0 s. He is driving on a circular track with a radius of 600. m.
Assuming constant tangential acceleration, calculate (a) tangential
acceleration, (b) centripetal acceleration at 40.0 m/s, and (c) Total
acceleration at 40.0 m/s.
|
|
/\v
v2
aT = -— and aC = —--
and a = ( aC2 + aT2
) 1/2
/\t
r
/\v
43.0 m s -1
aT = -— = ----------------- = 3.58 m s
-2
/\t
12.0 s
v2 (40.0 m s
-1 ) 2
aC
= —-- = ---------------------- = 2.67 m s -2
a = ( aC2 + aT2
) 1/2 = ( (3.58 m s -2)2
+ (2.67 m s -2)2 ) 1/2
= 4.47 m s -2
If the race is run counter-clockwise on the
track, the acceleration and force will be to the left of the tangential
direction
aC
2.67 m s -2
Angle = Tan -1( ------ ) = Tan -1 (
------------------ ) = 36.7o
aT
3.58 m s
-2
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| 2. |
Apollo went from rest to 20.0 m/s in 5.00
seconds on the ice skating short track at the 2002 Olympics with a radius
of 25.0 m. Assuming constant tangential acceleration, calculate (a)
tangential acceleration, (b) centripetal acceleration at 4.00 seconds, and
(c) Total acceleration at 4.00 seconds.
|
|
/\v
v2
aT = -— and aC = —--
and a = ( aC2 + aT2
) 1/2
/\t
r
/\v
20.0 m s -1
aT = -— = ----------------- = 4.00 m s
-2
/\t
5.00 s
v = at = 4.00 m s -2 x 4.00 s
= 16.0 m s -1
v2 (20.0 m s
-1 ) 2
aC
= —-- = ---------------------- = 16.0 m s -2
r
25.0 m
a = ( aC2 + aT2
) 1/2 = ( (4.00 m s -2)2
+ (16.0 m s -2)2 ) 1/2
= 16.5 m s -2
If the race is run counter-clockwise on the
track, the acceleration and force will be to the left of the tangential
direction
aC
16.0 m s -2
Angle = Tan -1( ------ ) = Tan -1 (
------------------ ) = 76.0o
aT
4.00 m s
-2
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