Aditya's challenges in Mechanics 5

Classical Mechanics Level 4

There is a fixed hemispherical cup of radius R = 1 m R=1 \text{ m} . There are 5 point sized balls each of mass m = 1 kg m=1 \text{ kg} randomly placed inside the cup and held there. At time zero, the balls are set free.

What is the maximum possible heat (in Joules) that can be produced by an arrangement of this system via collisions?

Details and Assumptions :

  • g = 10 ms − 2 g=10 \text{ ms}^{-2} .
  • Coefficient of restitution for each collision e = 0 e=0 .
  • This is a theoretical problem.

Try more here .


The answer is 50.

Aditya Kumar by Aditya Kumar , 22, India

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2 solutions

The balls are initially at rest. To extract the maximum amount of heat, the balls must start with maximum gravitational potential energy. Setting the reference at the bottom of the cup, maximum PE is achieved when the balls are at the edge of the cup. Q = 5 mgR = 50 J.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Avi Ghanshani
Mar 21, 2016

For maximum amount of heat to be liberated,1)potential energy should be maximum,2)they must collide simultaneously ,3) their net momentum at the lowest point must be zero.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you justify your second statement that they must collide simultaneously to produce maximum heat? As far as I can think, If we notice the point masses after some large enough time and if the system is isolated, the heat energy at the end will be 50J if we let them fall from maximum height no matter when and how they collide.

Anupam Nayak - 4 years, 11 months ago

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