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Simple Harmonic Motion

A humbling fraction of physics boils down to direct application of simple harmonic motion, the description of oscillating objects. Learn the basis for springs, strings, and quantum fields.

Level 3-4

         

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 \(T\) of this pendulum.

Note: Use \(\pi = 3.1416\) and \(g = 9.81\) m/s\(^2\).

A clock is made out of a disk of radius \(R = 10~\mbox{cm}\) 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 \(\frac{R}{2}\) 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 \(g = 9.81~\mbox{m/s}^2\)
  • 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 \(1\textrm{ kg}\) is dropped from a height \(H=2\textrm{ m}\). 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 \(100 \textrm{ Nm}|\).
    • If necessary, take the gravitational acceleration to be \(9.8\textrm{ m/s}^2\).

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 \(\eta=1.5 \) times, that is \[ \frac{T_{location 1}}{T_{location 2}}= 1.5. \] How much did the magnetic field of the earth change? In other words, determine \[ x=\frac{B_{location 2}}{B_{location 1}}.\] 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 \(T\) between successive bounces and the amplitude \(A\) 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: \(T\) proportional to \(A^w\). Find \(w\).

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