Objectives:
1. To verify Hooke's law for a linear spring, and
2. To verify the formula for the period, T, of an oscillating mass-spring system
Equipment:
A linear spring, slotted weights, a stop watch, a spring hanger, a meter stick or a 30.0-cm ruler, a mass scale, a C-clamp and rod attachment, a skew clamp, a regular weight hanger and a few sheets of Cartesian graph paper
Theory:
Hooke's law simply states that for a linear spring the spring force, F_{s }, is proportional to the change in length, x, that the spring undergoes. (Here, the term "spring force" means the force exerted by the spring on the object attached to it. The object is often called the "mass."
Mathematically, F_{s} = - kx, where k is the spring constant. The reason for the (-) sign is that F_{s} and x always have opposite signs. If a spring is pulled to the right, Fig. (a), the externally applied force, F_{appl.}, is to the right, but the spring force, F_{s} , acts to the left. On the other hand, when the spring is pushed to the left, Fig. (c), F_{s} acts to the right.
If mass M is hung from a spring as shown below, it stretches the spring of initial length y_{1} , and the spring attains an equilibrium length of y_{o} + y_{1}. If the mass is pushed up a distance A and then released, it oscillates above and below that equilibrium level. Distance A, that is the maximum deviation from equilibrium, is called the "amplitude" of oscillations.
This formula is a result of the solution to a 2^{nd} order linear differential equation with constant coefficients. The differential equation is set up very easily as follows. At any instant of oscillation, it is the spring force F_{s} = -ky that accelerates mass M at a rate a = d^{2}y /dt^{2} . According to Newton's 2^{nd} law, F_{s} = Ma. This may be written as:
- ky = Ma, or - ky = Md^{2}y /dt^{2}, or d^{2}y /dt^{2} + (k /M) y = 0.
This may be written as:
d^{2}y /dt^{2} + ω^{2} y = 0 where ω^{2} = k /M from which ω = ( k /M )^{(1/2)}.
Procedure:
The value of k, the spring constant, may be measured in two ways. One method is to use Hooke's law. The other method is to measure the period T of oscillations of a mass-spring system. The values of k determined by the two methods may then be compared and used as a verification of the validity of the theories involved.
I. The Hooke's Law Method:
The mass-spring system acts similar to a spring scale. It has a vertical ruler that measures the spring's elongation.
1. Measure the mass of the hanger without the spring.
2. Attach the spring and hanger to the support. Zero the system by sliding the ruler against the needle. The ruler slides easily once its collar or slider (at the back of the ruler) is squeezed with two fingers. By zeroing the system with its small weight hanger attached, you do not have to take its mass into account for this part of the experiment.
3. Place a 100g mass M_{1} on the hanger and measure the change in length of the spring Δy. It is better to use two 50g slotted masses instead of a single 100g mass. Make sure that the slots are exactly parallel and opposite to each other such that the weights hang perfectly vertical. If the slots are not opposite to each other, the weights hang tilted, making the needle tilted and causing an incorrect reading of the scale. Record the measured values of M and Δy, and the calculated value of F, in a table similar to the one shown below.
4. Repeat the above step for two or three additional values of mass up to about 250g. Again use smaller slotted weights with the slots configured to avoid tilting.
M (kg) |
F = Mg (N) |
Δy (m) |
5. Plot F versus Δy and find the slope of the graph. The spring constant k is equal to the slope.
II. The Oscillation Period Method:
1. Place the first recommended mass on the weight hanger.
2. Add the mass of the weight hanger to this mass and record it in the appropriate space in a table similar to the one shown below.
3. Pull the mass with its weight hanger down to about 2 to 3 cm below its equilibrium level and release. Start counting oscillations when the mass reaches either the highest or the lowest point. Start counting at zero while starting a stopwatch. The greater the number of oscillations, the more accurate is the measurement of the period. Count about 25 to 50 oscillations, and stop the watch. Record the total time in the table, and calculate the period T and T^{2}. Repeat this procedure for all recommended masses.
4. Plot T^{2} versus M, and find the slope of the graph. The spring constant k is given by k = (2π)^{2}/slope, an equation which can be obtained from ω = 2π/T. Calculate the spring constant.
5. Calculate a percent difference for the k values obtained by the two methods.
Note that the oscillating mass is not just the mass of the slotted weights in each case. In each calculation, the mass of the weight hanger must be taken into account.
M (kg) |
Total Time (s) |
T (s) |
T^{2} (s^{2}) |
Data:
Given:
M_{1} = 100. g M_{2} = 150. g M_{3} = 200. g M_{4} = 250. g
Measured:
Mass of Weight Hanger =
Calculations:
Perform the calculations and calculate k in Method II using k = (2π)^{2}/slope.
Comparison of the Results:
Calculate a percent difference using
Conclusion:
To be explained by students.
Discussion:
To be explained by students.
Questions:
1) Is the solution to the differential equation d^{2}y/dt^{2} + ω^{2} y = 0 of the form:
y = A cos (ω t) + B sin (ω t) ? If yes, what is the role of ω in the equation? What unit should it have if t is in seconds?
2) If a second order linear d.e. has the form: d^{2}y/dt^{2} + (k /M) y = 0, what should the value of ω be?
3) If ω is the angular frequency and in rad/s, how are f in (cycles/s) and T in (s) related to it?