Tutorial 5 - Circular Motion
Consider a car of mass m moving on a circular trajectory of radius on an inclined track.
The track is inclined at an angle with respect to the horizontal direction. Suppose the surface
of the inclined track is frictionless.
A. Draw the free-body diagram for the car. Indicate the direction in which the car is accelerating.
B. Define the direction of the car’s net force as the x-axis, and the direction perpendicular
to it as the y-axis.
C. Write down the x and y components of the Momentum Principle, ~
Fnet = m~a, from the
free-body diagram.
D. Calculate the speed at which the car must move to avoid sliding up or down the track.
Problem 2: Car on track with friction Consider a car of mass M moving on a circular
track at a radius R from the center of the track, which has a coefficient of kinetic friction µ .
The track is inclined at an angle θ with respect to the horizontal direction and the presence
of friction permits a range of speeds on the track. The car slides down the track for v < vmin
and slides up for v > vmax. (Make sure you understand why.)
Your task is to find the value for vmax
A. Draw the free-body diagram for the car, and indicate the direction in which the car is
accelerating. (Be careful about the direction of the frictional force.)
B. Define the direction of the car’s net force as the x-axis, and the direction perpendicular
to it as the y-axis.
C. Write down the x and y components of the Momentum Principle, ~
Fnet = m~a, from the
free-body diagram.
D. Obtain an algebraic expression for vmax in terms of R, g, µ and θ .
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Problem 3: Three body problem
You may be surprised to learn there is no general solution for the motion of a three-body
gravitational system. However, there are some special cases such as the one we will solve
today.
Imagine that the Earth, Mars and Venus all have the same
mass M, and are in circular motion with a radius R, remaining equidistant from each other, as shown in Fig. 1.
How long does this system takes to make one complete
R
revolution?
Figure 1: The Earth, Mars and Venus were captured by
Darth Vader and made to rotate in a circle of radius R
equidistant from each other.
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