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Solved: In a lab experiment you let a uniform ball roll
Chapter 10, Problem 85P(choose chapter or problem)
In a lab experiment you let a uniform ball roll down a curved track. The ball starts from rest and rolls without slipping. While on the track, the ball descends a vertical distance h. The lower end of the track is horizontal and extends over the edge of the lab table; the ball leaves the track traveling horizontally. While free-falling after leaving the track, the ball moves a horizontal distance x and a vertical distance y. (a) Calculate x in terms of h and y, ignoring the work done by friction. (b) Would the answer to part (a) be any different on the moon? (c) Although you do the experiment very carefully, your measured value of x is consistently a bit smaller than the value calculated in part (a). Why? (d) What would x be for the same h and y as in part (a) if you let a silver dollar roll down the track? You can ignore the work done by friction.
Questions & Answers
QUESTION:
In a lab experiment you let a uniform ball roll down a curved track. The ball starts from rest and rolls without slipping. While on the track, the ball descends a vertical distance h. The lower end of the track is horizontal and extends over the edge of the lab table; the ball leaves the track traveling horizontally. While free-falling after leaving the track, the ball moves a horizontal distance x and a vertical distance y. (a) Calculate x in terms of h and y, ignoring the work done by friction. (b) Would the answer to part (a) be any different on the moon? (c) Although you do the experiment very carefully, your measured value of x is consistently a bit smaller than the value calculated in part (a). Why? (d) What would x be for the same h and y as in part (a) if you let a silver dollar roll down the track? You can ignore the work done by friction.
ANSWER:Solution 1
Introduction
First we have to calculate the horizontal velocity of the ball by using the conservation of energy. Then we have to calculate the time taken for free fall for distance. Then we can calculate the horizontal distance
in from the horizontal velocity.
(a)
Step 1
Suppose the mass of the ball is
The potential energy at the height is given by
Now let us consider that the ball is coming with velocity at the bottom of the track. Hence the kinetic energy of the ball is given by
Now, since the ball is rolling without slipping, we can write that the angular velocity is given by
Also for the moment of inertia for the ball is
Hence the kinetic energy becomes
Now equating the final kinetic energy with the initial potential energy in the track we have
Now after the table, this will be the x-component of the velocity.