Expert: Steve Johnson - 2/7/2009

Question
If a person weighing 100 lbs. started juggling three balls weighing 5 pounds each, stepped on (juggling had already begun) scales, juggled for 5 min, then stepped off. What would be the average weight on the scales for the five minutes?

Answer
Hello George,

This is an interesting question! Clearly if the person just stood there holding one of the balls, the scale should read 105 lbs. To throw that one ball up requires exerting more force, while throwing it, than just the 5 lbs to hold it. The amplitude of the force momentarily applied upward on the ball will be reflected as an increase over the basic 105 lb scale reading because the juggler needs to remain upright. While the ball is in the air, the scale should return to 100 lbs.

That ball is staying above the floor. It has a gravitational attraction to the earth. Without something supporting it, it would fall to the floor. The fact that it's going up, down, up, down doesn't change the fact that it is being kept off the floor. Consider a 5 lb rocket being controlled in a way that it hovers. The rocket must be applying 5 lbs of force on it. What if the control isn't perfect so it can't just hover -- what if it goes up, down, up, down; varying for a period of 5 minutes between 1 meter and 10 meters above the ground, kind of as if I were at the controls. The average force still must be 5 lbs. So it would be with a juggler and a single 5 lb ball.

Now the question is what change is there if there are 3 balls. There are different styles of juggling. It can be done so the juggler deals with only one ball at a time. So can it be that the average scale reading is just 105 lbs? No, if 3 balls are being kept off the floor, it must be 115 lbs. That's my intuitive attack on the problem.

Now a little more scientific analysis using the relationship between impulse and momentum.
Change in momentum = impulse.
Impulse is net force applied times the time duration of the application of this net force. If the balls are just held for 5 minutes, there is no net force acting on the balls. The gravitational attraction is 15 lbs down and the juggler is applying upward support of 15 lbs to them. And the scale has to support that extra 15 lbs. Regarding the balls, Fnet = 0. Once juggling starts, they are going up, down, up, down, but you could say that their average momentum is zero. Because they stay in the area above the juggler's hands. Using the relationship between impulse and momentum, if average momentum remains zero, average impulse is zero. Time is not zero, so the average net force is zero. So on average, the juggler, and therefore the scale, provides 15 lbs support to the balls, this force being equal and opposite to the balls' weight.

I hope this helps,
Steve