## Question:

Newton’s cradle is a common desk toy consisting of a number of identical pendulums with steel balls as bobs. These pendulums are arranged in a row in such a way that, when at rest, each ball is tangent to the next and the cords are all vertical. Assume that the COR for the impact of a ball with the next is e = 1. Predict the swing pattern of the system in the figure if you release from rest, and from a given angle, three of the five balls.

## Step-by-step

Treat each impact as only involving two balls. The first impact will occur between balls 3 and 4. Because the COR e = 1 and the masses are identical we see from the solution to Problem 5.47 that ball 3 will come to a complete stop after impacting with ball 4. We also see that ball 4 will have a post impact velocity identical to the pre impact velocity of ball 3. Call this velocity ${ \upsilon }_{ 0 }$. At the same instant ball 3 impacts ball 4 ball 2 impacts ball 3. Ball 2 will stop and ball 3 will have a post impact velocity ${ \upsilon }_{ 0 }$. Also at the same instant ball 1 will impact ball 2. Ball 1 will stop and ball 2 will have a post impact velocity ${ \upsilon }_{ 0 }$. Now balls 2, 3, and 4 have velocity ${ \upsilon }_{ 0 }$. When ball 4 impacts ball 5 ball 4 stops and ball 5 has a post impact velocity ${ \upsilon }_{ 0 }$, ball 4 is impacted by ball 3, ball 3 stops and ball 4 has a post impact velocity ${ \upsilon }_{ 0 }$. At the same instant ball 2 impacts ball 3. Ball 2 stops and ball 3 has a post impact velocity ${ \upsilon }_{ 0 }$. Now balls 3, 4, and 5 have velocity ${ \upsilon }_{ 0 }$ and there are no other balls in the train to impact so they will all swing together as a unit. The work-energy principle tells us that balls 3, 4, and 5 will stop moving when they have reached the initial height balls 1, 2, and 3 were released from. Finally, since the lengths of the pendulums are identical the maximum swing angle of balls 3, 4, and 5 are equal to the initial release angle of balls 1, 2, and 3.

Balls 3, 4, and 5 will swing up as a single unit until reaching the height that
balls 1, 2, and 3 were released from while balls 1 and 2 will hang motionless.