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Name: Brian J.
Status: other
Age: 50s
Location: N/A
Country: N/A
Date: 3/19/2003

(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

Dear Brian,

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.

Martha Croll


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
Physics Instructor
Illinois Central College

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