Classical Mechanics

Simple Harmonic Motion

Simple Harmonic Motion: Level 3-4 Challenges


An inventor designs a pendulum clock using a bob with mass 200 g at the end of a thin wire of length 23 cm. Instead of swinging back and forth, the bob is to move in a horizontal circle, making a fixed angle 27° with the vertical. This is called a conical pendulum because the suspending wire traces out a cone. Find the period TT of this pendulum.

Note: Use π=3.1416\pi = 3.1416 and g=9.81g = 9.81 m/s2^2.

A clock is made out of a disk of radius R=10 cmR = 10~\mbox{cm} which is hung by a point on its edge and oscillates. All of a sudden, a circular part right next to the hanging point of radius R2\frac{R}{2} falls off, but the clock continues oscillating. What is the absolute value of the difference in s between the periods of oscillation before and after the part fell off?

Details and assumptions

  • Gravitational acceleration is g=9.81 m/s2g = 9.81~\mbox{m/s}^2
  • Amplitude of the vertical oscillations is small
  • The axis of rotation of the disc is horizontal all the time
  • The disc is homogeneous

A ball of mass 1 kg1\textrm{ kg} is dropped from a height H=2 mH=2\textrm{ m}. Estimate how long the ball is in contact with the ground (in seconds)?

Details and assumptions

  • Simplify the question by modeling the ball as an ideal spring of spring constant 100 Nm100 \textrm{ Nm}|.
    • If necessary, take the gravitational acceleration to be 9.8 m/s29.8\textrm{ m/s}^2.

A small magnetic needle in a compass performs small oscillations about an axis perpendicular to the Earth's magnetic induction field. On a different Earth location it is observed that the needle's oscillation period decreased by η=1.5\eta=1.5 times, that is Tlocation1Tlocation2=1.5. \frac{T_{location 1}}{T_{location 2}}= 1.5. How much did the magnetic field of the earth change? In other words, determine x=Blocation2Blocation1. x=\frac{B_{location 2}}{B_{location 1}}. You may neglect the Earth's gravitational field in this problem.

Not every oscillation in nature is a harmonic oscillation - in this problem, we will examine a non-harmonic oscillation. Suppose we had a rubber ball with a perfect coefficient of restitution so that, when dropped, it would always return to the same height. The period of the bouncing is the time TT between successive bounces and the amplitude AA of this non-linear oscillation is the maximum height of the ball above the floor. For a harmonic oscillator, the period and amplitude are independent, but not so here. Instead, there's a relation between them: TT proportional to AwA^w. Find ww.


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