Determination of Force Constant k from Spring-mass System

 

We can see that the mass will accelerate upward.

As the mass is moving in SHM,  (by SHM equation)

So by measuring the oscillation period and the mass of mass, we can calculate the spring constant (or say the force constant).

Apparatus:

Slotted mass (10g, 20g, and 50g), hanger, spring, retort stand and clamp, stop watch.

Procedure:

1. Fix the spring to the clamp with the help of an eraser. Hook the spring to the eraser and place the eraser in the clamp and screw up the clamp.
2. Hang the hanger which has a mass of 20g to the spring.
3. Use the G-clamp to fix the stand on the table. So when the experiment starts, the support is well fixed.
4. Put a slotted mass on the hanger and weight the total mass.
5. Pull the slotted mass and the hanger down for around 2cm and release it and make the oscillations as vertical as possible.
6. Try a few trial oscillations and ensure the oscillation is vertical.
7. Start the stopwatch and take the time t for 20 oscillations.
8. To reduce the random error, for each time adding one more slotted mass, take 3 sets of data for the period.
9. Calculate the time for one oscillation from the mean value of time for 20 oscillations. Plot a graph of T2 against m by Microcal Origin 6.0. Fit the points by the best straight line passing through the origin as if there is no mass hanged by the spring, there will not be any oscillation.
10. From the software, the slope M of the best fit line is calculated with error. The force constant k can be calculated by the equation:

Results:

The values of mass, time t for 20 oscillations and period square T2 (one oscillation) are recorded below:

Let  be the mean value of t

T2 can be calculated be the equation: 

The graph of T2 against m is plotted as shown below:

A graph of T2 against m of a spring-mass system

The slope of the graph is calculated by the software as M=11.714150.04899 s2kg-1.

By the equation  and, the force constant k is calculated as

k=3.370.01Nm-1

Discussion and analysis:

The % error of the spring constant value k is

However, there is not theoretical value provided.

The value k has errors due to the following reasons:

1. The mass is not accurate. Initially, the mass of all slotted masses should be 10g, but it is actually not. All of them are below 10g so we have to measure the mass and cause inaccuracy.
2. There is an external force throughout the oscillation but the equation is estimating there is not external force. The air resistance F =bv which oppose the motion of the masses. So overall, the system does positive work on the surrounding and energy is taken away from the oscillations. This makes the period greater and so the force constant smaller. If the amplitude is too large, the damping force bv is than greater than the restoring force. This causes a much longer period.
3. As the slotted masses can be separated, during the oscillations, all slotted masses have inertia towards the two side end. When the object (masses and hanger) arrive to the highest position, it is just like projecting an object upward and it is free falling. This causes a larger force pressing downward and so a smaller T deduced.
4. Even though we use the G-clamp to fix the stand on the table, the table itself is still movable; the stick of the stand is movable also. So using G-clamp cannot assure that the system is not oscillating about one point.
5. Overall, the locus of the masses is not a vertical line but a circle, so the actual storing force in spring is only partly cancelled out by weight.