Acceleration of Falling Balls on Elastic Bands

In summary, two small balls with masses 3M and M are suspended on elastic rubber bands. When the band between the balls is cut, the M ball falls downward at an acceleration of 9.8 m/s^2, while the 3M ball moves upwards at an acceleration of 1/3 g or 3.27 m/s^2. This is due to the release of tension in the top rubber band, which results in a net upward force of 4M on the 3M ball.
  • #1
grouper
52
0

Homework Statement



Two small balls with masses 3M and M hang on elastic massless rubber bands (with the M ball attached to and suspended beneath the 3M ball with one rubber band; the 3M ball is suspended from the ceiling by a separate rubber band). When the band between the balls is cut, what are the accelerations of the balls immediately after?

Homework Equations



F(GRAVITY)=mg

The Attempt at a Solution



I think the M ball would simply move downward at acceleration g=9.8 m/s^2. But would the 3M ball move upwards due to the release of the tension when the rubber band is cut? And would that upward acceleration be equal to the force of gravity on the M ball? How do you take the force of gravity on the 3M ball into account?
 
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  • #2
Why would the rubber band have any effect on the ball after it's cut? The rubber band would snap up & away from the ball to relieve the tension, and gravity would be the only force on the ball.
 
  • #3
What is the upward force on the 3M ball before cutting the lower rubber band?
This force will act on the ball right after cutting the band. There is also a downward force that is balanced by the upward force before cutting the band but won't be balanced anymore after cutting.
 
  • #4
The nature of the rubber bands is not explained and I don't think relevant; I gave all the information given in the problem. The 3M ball (on top) has the downward force of gravity and the downward tension from the M ball (below it) and the upward tension from the top rubber band (the tension being from the masses of both of the balls, so 4M) initially, correct? So when you remove the M ball on the bottom, I believe the 3M ball has just the downward force of gravity from its own mass and the upward 4M tension from the rubber band. (Please correct if I'm wrong about this). So the M ball will fall normally, but the 3M ball will accelerate upwards due to the 4M tension. But how fast is that?
 
  • #5
Oh, I didn't read the question properly. Apologies.

What's the net force on the 3M ball if there's an upwards force of 4M and a downwards force of 3M? (Of course, it's actually 4Mg and 3Mg.) Fnet=ma will give you the answer.
 
  • #6
So the M ball falls at 9.8 m/s^2 and the 3M ball falls at 1/3 g (or 3.27 m/s^2). Did I understand you correctly?
 
  • #7
The 3M ball does not fall but moves upwards (right after cutting the lower band).
 
  • #8
Of course, sorry. That's what I meant. Thanks for the help!
 

1. What is the acceleration of a falling ball?

The acceleration of a falling ball is the rate at which the ball's velocity changes over time due to the force of gravity. On Earth, the acceleration of a falling ball is approximately 9.8 meters per second squared.

2. Does the mass of the ball affect its acceleration?

Yes, the mass of the ball does affect its acceleration. According to Newton's Second Law of Motion, the acceleration of an object is directly proportional to the force applied to it and inversely proportional to its mass. This means that a larger mass will have a smaller acceleration compared to a smaller mass under the same force of gravity.

3. How does air resistance affect the acceleration of a falling ball?

Air resistance, or drag, is a force that opposes the motion of an object through the air. As a falling ball accelerates, the force of air resistance also increases, causing the ball to reach a terminal velocity where the force of air resistance equals the force of gravity. This means that the ball will no longer accelerate and will fall at a constant speed.

4. What is the difference between acceleration due to gravity and acceleration of a falling ball?

Acceleration due to gravity refers to the acceleration of any object near the surface of the Earth due to the force of gravity. The acceleration of a falling ball specifically refers to the acceleration of a ball as it falls towards the ground due to the force of gravity.

5. How can the acceleration of falling balls be calculated?

The acceleration of falling balls can be calculated using the equation a = g - (Fd/m), where a is the acceleration, g is the acceleration due to gravity, Fd is the force of air resistance, and m is the mass of the ball. This equation takes into account the effects of air resistance on the acceleration of the ball.

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