Physics Help. A ball of mass 540 g hangs from a spring whose stiffness is 120 newtons per meter.

Question

A string is attached to the ball and you are pulling the string to the right, so that the ball hangs motionless, as shown in the figure. In this situation, the spring is stretched, and its length is 15 cm (Hypotenuse). The base is 8cm.

 

What would be the relaxed length of the spring, if it were detached from the ball and laid on a table?

Answer 1

Hooke's law for a spring is

 

F=k*x

 

where F is the force stretching the spring, k is the spring constant, and x is the displacement of the spring.

 

Any information regarding the length of the spring is given by the value 15cm, which is the original length of the spring plus the displacement due to applied forces. Since we know k = 120 N/m, we have the force and the displacement to figure out.

 

The problem states that the force on the spring when free-hanging comes from the ball of mass 540g, or F_y=(0.54 kg)*(9.81 m/s²) = 5.2974 N. This force is due to gravity and pulls the spring downward.

 

Then you pull the ball to the right, exerting a purely horizontal force on the ball and thus the spring. We don't care so much about this horizontal force. We care about the force on the spring since we have that spring length information. So, in order to find the force on the spring, which is along the hypotenuse, we need to know the angle at which this force is being applied.

 

We are given the length of the spring and the base length which gives us two sides to a right triangle which we can use to find our force. So, let's find the third side using Pythagorean theorem.

 

15² = 8² + y²

y = 12.689

 

Now, use ratios to find the force on the spring. In this case, we do the y-component over the hypotenuse.

 

12.689/15 = 5.2974/F

F = 6.262 N

 

Now use Hooke's Law to find the displacement.

 

F=k*x

6.262 = 120*x

x = 5.2186 cm

 

This is the displacement of the spring at the angled configuration. We know the full length of the spring is 15cm, so then

 

15 - 5.2186 = 9.7813 cm

 

So, 9.78 cm is your relaxed length of the spring.

 

Note: You could find the angle and use sine and cosine to calculate your values, but I find it easier to not mess with them if I don't have to.