The oscillating object does not lose any energy in SHM. Friction is assumed to be zero.

In harmonic motion there is always a *restorative force,* which attempts to *restore* the oscillating object to its equilibrium position. The restorative force changes during an oscillation and depends on the position of the object. In a spring the force is given by Hooke’s Law: \begin{align*}F = -kx\end{align*}

The period, \begin{align*}T\end{align*}

The frequency, \begin{align*}f,\end{align*}*equilibrium* (or center) *point* of motion to either its lowest or highest point (*end points*). The amplitude, therefore, is half of the total distance covered by the oscillating object. The amplitude can vary in harmonic motion, but is constant in SHM.

\begin{align*} T = \frac{1}{f}\end{align*}

\begin{align*} T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}}\end{align*} ; Period of mass m on a spring with constant k

\begin{align*} F_{sp} = -kx \end{align*} ; the force of a spring equals the spring constant multiplied by the amount the spring is stretched or compressed from its equilibrium point. The negative sign indicates it is a restoring force (i.e. direction of the force is opposite its displacement from equilibrium position.

\begin{align*} U_{sp} = \frac{1}{2} kx^2 \end{align*} ; the potential energy of a spring is equal to one half times the spring constant times the distance squared that it is stretched or compressed from equilibrium

### Review

- A rope can be considered as a spring with a very high spring constant \begin{align*}k,\end{align*}so high, in fact, that you don’t notice the rope stretch at all before it “pulls back.”
- What is the \begin{align*}k\end{align*} of a rope that stretches by \begin{align*}1\;\mathrm{mm}\end{align*} when a \begin{align*}100\;\mathrm{kg}\end{align*} weight hangs from it?
- If a boy of \begin{align*}50\;\mathrm{kg}\end{align*} hangs from the rope, how far will it stretch?
- If the boy kicks himself up a bit, and then is bouncing up and down ever so slightly, what is his frequency of oscillation? Would he notice this oscillation? If so, how? If not, why not?

- If a \begin{align*}5.0\;\mathrm{kg}\end{align*} mass attached to a spring oscillates 4.0 times every second, what is the spring constant \begin{align*}k\end{align*} of the spring?
- A horizontal spring attached to the wall is attached to a block of wood on the other end. All this is sitting on a frictionless surface. The spring is compressed \begin{align*}0.3\;\mathrm{m}\end{align*}. Due to the compression there is \begin{align*}5.0\;\mathrm{J}\end{align*} of energy stored in the spring. The spring is then released. The block of wood experiences a maximum speed of \begin{align*}25\;\mathrm{m/s}\end{align*}.
- Find the value of the spring constant.
- Find the mass of the block of wood.
- What is the equation that describes the position of the mass?
- What is the equation that describes the speed of the mass?
- Draw three complete cycles of the block’s oscillatory motion on an \begin{align*}x\end{align*} vs. \begin{align*}t\end{align*} graph.

- A spider of \begin{align*}0.5\;\mathrm{g}\end{align*} walks to the middle of her web. The web sinks by \begin{align*}1.0\;\mathrm{mm}\end{align*}due to her weight. You may assume the mass of the web is negligible.
- If a small burst of wind sets her in motion, with what frequency will she oscillate?
- How many times will she go up and down in one s? In \begin{align*}20\;\mathrm{s}\end{align*}?
- How long is each cycle?
- Draw the \begin{align*}x\end{align*} vs \begin{align*}t\end{align*} graph of three cycles, assuming the spider is at its highest point in the cycle at \begin{align*}t=0\;\mathrm{s}\end{align*}.

### Review (Answers)

- a. \begin{align*}9.8 \times 10^5 \;\mathrm{N/m}\end{align*} b. \begin{align*}0.5 \;\mathrm{mm}\end{align*} c. \begin{align*}22 \;\mathrm{Hz}\end{align*}
- \begin{align*}3.2 \times 10^3 \;\mathrm{N/m}\end{align*}
- a. \begin{align*}110 \;\mathrm{N/m}\end{align*} b. 0.016 kg c. \begin{align*}0.3mcos(82t)\end{align*} d. \begin{align*}v(t)=(25) \cos(83\mathrm{t})\end{align*}
- a. \begin{align*}16 \;\mathrm{Hz}\end{align*} b. \begin{align*}16\end{align*} complete cycles but \begin{align*}32\end{align*} times up and down, \begin{align*}315\end{align*} complete cycles but \begin{align*}630\end{align*} times up and down c. \begin{align*}0.063 \;\mathrm{s}\end{align*}