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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.

# 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 $$T$$ 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 coefficient 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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