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.
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 y distance. Then we can calculate the horizontal distance x in from the horizontal velocity. (a) Step 1 Suppose the mass of the ball is m The potential energy at the height h is given by U = mgh Now let us consider that the ball is coming with velocity v 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. Step 2 Now we have to calculate the time taken by the ball to fall freely for y distance. From the gravity, we know the time taken for free fall for the distance y is given by Step 3 Now the x distance is the distance travelled by the ball in the above calculated time. Now since the horizontal velocity of the ball is v , the distance travelled is given by (b)