Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1310003     A block hanging vertically from a spring in equilibrium.     1310003
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Question

 

In the lab, you connect one end of a 15 N/m spring to a ringstand, and you hang a 0.25-kg block from the other end of the spring. It hangs vertically in equilibrium.

  1. How far is the spring stretched from its relaxed length?
  2. Suppose that you put your hand beneath the block and lift up on the block in order to partially support it. It is still in equilibrium with the spring stretched 5 cm from its relaxed length. What is the force of your hand on the block?

 



Solution

 

Begin by drawing a picture, showing the spring and block. Indicate the stretch of the spring.


Figure: A block hanging in equilibrium from a spring.

(a) Apply the momentum principle and note that the block's momentum is not changing (after all, it's at rest). This is what it means to be in equilibrium.

Sketch a free-body diagram showing the forces acting on the block.


Figure: A free-body diagram for the block.

Since the block is in equilibrium, the net force on the block is zero. Now, apply the definition of net force as the sum of the forces acting on the block to calculate the net force. Use Hooke's law for the spring force.

Write the above equation in component form for the y-component, and solve for the distance stretched.

The spring stretches 0.16 m (16 cm) from its relaxed length. If you've done experiments in the classroom with typical springs used for Hooke's law experiments, then you'll recognize that this distance is reasonable.

(b) Now, you push up on the block with your hand. The spring is stretched 5 cm. Draw a picture of the situation.


Figure: Your hand pushes up on the block.

Draw a free-body diagram for the block. The sum of the upward forces must be equal in magnitude to the downward force since the net force on the block is zero (because it is in equilibrium and its momentum is constant, i.e. it is at rest).


Figure: A free-body diagram for the block when partially supported by your hand.

Again, apply the definition of net force by summing the forces. Note that because of the momentum principle, the net force is zero. Solve for the force by the hand on the block.

Check the direction. It is upward, in agreement with the free-body diagram. It is also less than the gravitational force on the block (2.45 N), as expected. And since the spring is stretched less than half its original distance of 0.16 m, one would expect the force by the hand on the block (1.7 N) to be greater than the force by the spring on the block (0.75 N) which it is. You can therefore be confident that the answer is correct.