Tutorial 5 - Circular Motion
PHY 303K
Problem 1: Car on Inclined Track without Friction
Consider a car of mass M moving with a constant speed v on an inclined circular track. The
track is inclined at an angle with respect to the horizontal direction. The car’s trajectory
on the track is uniformly circular at a …xed height with radius R: Suppose the surface of
the track is frictionless. You task is to calculate v.
1. Draw the free-body diagram for the car and separately indicate the direction in which
the car is accelerating.
2. De…ne the direction of the car’s net force as the x-axis, and the direction perpendicular
to it as the y-axis.
!
3. Write down the x and y components of the Momentum Principle, F net = m!
a ; from
the free-body diagram.
4. Obtain an algebraic expression for v in terms of R; g; and ; where g is the magnitude
of acceleration due to gravity.
Problem 2: Car on Inclined Track with Friction
Consider a car of mass M moving with a constant speed v on an inclined circular track. The
track is inclined at an angle with respect to the horizontal direction. The car’s trajectory
on the track is uniformly circular at a …xed height with radius R: Suppose the surface of the
track has a coe¢ cient of static friction s : The presence of friction allows 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 calculate vmax :
1. Draw the free-body diagram for the car, and separately indicate the direction in which
the car is accelerating. (Be careful about the direction of the frictional force; you are
calculating vmax )
2. De…ne the direction of the car’s net force as the x-axis, and the direction perpendicular
to it as the y-axis.
!
3. Write down the x and y components of the Momentum Principle, F net = m!
a ; from
the free-body diagram.
4. Obtain an algebraic expression for vmax in terms of R; g;
magnitude of acceleration due to gravity.
s;
and ; where g is the
5. Consider: Are there any angles for which no vmax exists? Why or why not?
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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 radius R, remaining equidistant from each other as shown in Fig.(1) :
Figure 1: Three-Body Gravitational System
Question: How long does this system take to make one complete revolution?
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