Stevens High School AP Physics I Laboratory Manual

Purpose: The purpose of this laboratory is to show you how the mass of an object can be deduced without weighing it and to give you a little experience with spring stiffness (often denoted ‘k’)

Concept: Hooke’s Law gives a relationship between the force that an ideal spring exerts (Fs) when it is displaced by ‘x’ from equilibrium (the point where it is neither stretched nor compressed):

or

Here, k is the spring constant (stiffness) of the spring, with units of N/m. The minus sign is very important here! As you move further into physics, equations of this form will be more common, and the sign denotes the fact that the spring force is a restoring force. The minus sign tells you that the force always acts to restore the spring to the equilibrium position. Furthermore, equation (1) indicates that the relationship between force and displacement should be linear. Many springs will be linear for most of their range, but diverge at large extensions or compressions. In other cases, special springs are manufactured to have a spring constant that varies with displacement.

If a spring is displaced from equilibrium and then released, it will oscillate back and forth in simple harmonic motion (neglecting losses due to friction and internal stresses) forever. The interesting part is that, for an ideal spring following equation (1), the period of this motion is independent of the amount you stretch the spring before releasing it. Here, period is denoted with the symbol T, and is the time required for one complete oscillation:

This equation provides a useful way to do two things: It can be used to measure the spring constant of a spring accurately to high precision, and it gives you a way to determine mass m without weighing an object. You will do both in this laboratory.

Materials:

  • A spring with low stiffness
  • A set of masses
  • An unknown mass
  • A board and clamp
  • Writing substrate and utensil.

Procedure:

  1. Hang your spring off of the edge of the laboratory table using the board and clamps.
  2. Hang a known mass off of the spring, being careful not to overstretch it. This is called plastic deformation, and it will ruin the spring (and your grade).
  3. Prepare a stop watch.
  4. Next, displace the weight a small amount and release it the moment the stop watch is started.
  5. Measure the time required for 20 oscillations, and divide this time 20 to find the period of oscillation. Record it in a table resembling the one below.
  6. Repeat steps 2-5 with at least 6 different masses.
  7. Next, obtain an unknown mass from Dr. Smith.
  8. Repeat steps 2-5 with the unknown mass, and enter the period in the table.
  9. Weigh the unknown mass, and record this value.

Mass (kg) / Period (s)
Unknown

Table 1: Raw data from the inertial mass lab

Analysis:

  1. Linearize equation (2), giving this result:

Note that now you can plot the period squared as your y-axis and mass as your x-axis and your line should have a slope of . Do so, add a best fit line with an equation, and find the spring constant for your spring.

  1. Next, determine the mass of the unknown mass using your plot and the measured period.
  2. Calculate the percent error between the mass you determined using the spring oscillation period and using the balance.

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