Unit 12: Springs & Simple Harmonic Motion
Practice Problems
Note: problem difficulty is ranked using a star system.
(*) One-star problems are fundamental to the unit, and can be done relatively quickly. Use these problems to introduce yourself to the material.
(**) Two-star problems are more difficult, and require an understanding of one or two key concepts. Use these problems to test your understanding of the material.
(***) Three-star problems are the most difficult, and require some creative thinking in addition to a deep familiarity with multiple key concepts. Use these problems to challenge yourself; if you can complete one of these, you’re on your way to mastering the material.
*Q12.1) An 80g mass rests on a frictionless horizontal surface and is attached to the end of a spring of force constant 1.52 N/m. After being pulled from its equilibrium position and released, it oscillates back and forth undergoing simple harmonic motion. What’s the frequency of the oscillation?
*Q12.2) How much energy is required to elongate the spring from Q12.1 by 6cm from its equilibrium length?
*Q12.3) Consider again the simple harmonic oscillator from Q12.1. Assuming the mass oscillates with an amplitude of 6cm and reaches maximum extension at t = 0,
a) Write the equation that describes the position of the mass as a function of time.
b) Sketch a plot of the function from part a).
*Q12.4) A 78kg mass is hung on a vertical spring, causing it to stretch a length of 2.5m. What’s the force constant of the spring?
**Q12.5) You pull downward on the mass from Q12.4, causing the spring to stretch by an additional 50cm before releasing it.
a) How long does it take for the mass to pass through the equilibrium position after being initially released?
b) How fast is it moving when it passes through the equilibrium position?
**Q12.6) A 10kg mass attached to a spring oscillates back and forth 4 times per second, undergoing simple harmonic motion with an amplitude of 20cm.
a) Find the speed of the mass when it passes through equilibrium.
b) Find the speed of the mass when it is 6cm from equilibrium.
c) Find the position of the mass at t = 3.2 seconds, assuming the mass was released from maximum extension at t = 0.
*Q12.7) You construct a simple pendulum by attaching a 7kg mass to the end of a string of length 40cm and of negligible mass. You raise the mass up through an angular distance of 20 degrees and release it, causing it to oscillate back and forth undergoing simple harmonic motion. How many oscillations does the mass complete in 6.35 seconds?
**Q12.8) What happens to the period of the oscillating pendulum from Q12.8 if:
a) the mass is doubled?
b) the amplitude is doubled?
c) the length of the string is quadrupled?
d) the experiment is performed on the moon (where g = 1.625m/s²)?
**Q12.9) You want to make a simple timer by constructing a pendulum that has a period of precisely one second. What length of string will accomplish this?