Motion and Mass on a Track
Name: Brian J.
(2) equal-size metal spheres, (1) aluminum, (1) gold, and a long,
straight track with a U-curve at the end.
(ignore friction; gravity will not matter either because the track is
either horizontal if on earth, or is in space)
If each sphere in turn is pushed at the same speed along the track, will
the U-turn cause them to have different speeds?
If I understand your question properly, the force of gravity is
perpendicular to the balls motion and the force of the track on the ball
is also perpendicular to the ball's motion, there being no friction.
Therefore, no work is done on the ball and so the kinetic energy of the
ball does not change. The speed of each ball will be unchanged as it
rolls down the track and goes around the U-curve.
This is based on the work-energy theorem. Newton showed, based on his
second law (F = ma) that the work done on an object equals its change in
kinetic energy. W = F x cos(theta) where F is the total force acting on
the object, x is the displacement of the object while that force is
acting, and theta is the angle between the force and the direction of
motion of the object. Kinetic energy is, of course, 1/2 m v^2. The work
done on the ball in your question is zero since theta is 90 degrees and so
the kinetic energy is constant, which means the velocity v is constant.
Best, Dick Plano Professor of Physics emeritus Rutgers University
Here is the scoop. The u-turn is not what is going to cause them to have
different speeds. If the spheres are equal size they will have a mass
difference because the density of aluminum is less than gold. Given the
frictionless, gravity-starved track, the only thing that will determine
speed will be due to the force imposed upon them. If forces are equal at
the outset, the gold sphere will have a smaller acceleration than the
aluminum sphere because of the density difference.
Now if you mean that the two spheres are actually pushed until they
achieve the same SPEED and then are released, I still do not think it
will change the outcome. If the track does not give, or become deformed
in any way as the spheres round the bend, the track should exert just
enough force to "turn" the spheres. This I think would be in proportion
to the spheres mass, so would not effect speed.
This is my best guess answer to your hypothetical question. Perhaps
someone else may see this differently.
If there is no friction and if the track is held stationary, the U-curve
will not change the speed of either ball. Without friction, the U-curve
always pushes exactly sideways. To speed up an object, there must be some
forward force. To slow down an object, there must be some backward force,
often provided by friction. At any given moment, the U-curve pushes
neither forward or backward. Velocity remains constant.
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012