# Block, spring, and mass problem

1. Aug 15, 2007

### kiwikahuna

1. The problem statement, all variables and given/known data
A 0.1 kg block is suspended from a spring. When a small stone of mass 28 g is placed on the block, the spring stretches an additional 4.4 cm. With the stone on the block, the spring oscillates with an amplitude of 14 cm.
(a) What is the frequency of the motion?
(b) How long does the block take to travel from its lowest point to its highest point?
(c) What is the net force on the stone when it is at a point of maximum upward displacement?

3. The attempt at a solution
I was able to solve a & b, but am having trouble with c.

First, I found the spring constant by using the info that the 28 g stretched the spring 4.4 cm.

F = k Δx or k = F / Δx = mg / Δx = 0.028 * 9.8 / 0.044 = 6.236 N/m

Then I found the angular freq of the motion:

ω = ( k / m )1/2 = ( 6.236 / 0.128 )1/2 = 6.98 rad/s

and then the freq is

f = ω / 2π = 6.98 / 2 π = 1.11 Hz

The period of the motion is

T = 1/f = 1 / 1.11 Hz = 0.900 sec so the time to get just from bottom to top is half this or

0.450 sec

Finally, I thought that the net force on the block at the top is

F = kA = 6.236 N/m * 0.14 m = 0.873 Newtons but this answer isn't right. What did I do wrong though?

Last edited: Aug 15, 2007
2. Aug 15, 2007

### dimensionless

I think your spring constant is wrong. With only the 0.1 kg block on the spring the spring stretches by an amount
$$\Delta x = \frac{-m_{1} g}{k}$$
where $$m_{1}$$ is the mass of the block. When the stone is placed on the block the spring stretches an additional 4.4 cm. This would give

$$0.044(meters) + \Delta x = \frac{-(m_{1}+m_{2}) g}{k}$$

Where $$m_{2}$$ is the mass of the stone. I would use these two equations to solve for $$k$$.

When both masses are on the spring, this would lead to the following equation of motion:

$$(m_{1}+m_{2}) \ddot{x} = -k x - (m_{1}+m_{2}) g$$

Where $$x$$ is the displacement from equilibrium when there is zero mass on the spring.

Last edited: Aug 15, 2007
3. Aug 15, 2007

### Dick

Your spring constant is just fine. The question asks for the net force on the stone, not on the block. It may help to find the acceleration of the block (and hence the stone) at the point.