Assignment 7

due Friday, Mar. 8

3.8.2 Solution

3.8.9 Solution

3.8.13 Solution

    

Note: In 3.8.9, you may want to convert to mks units: 1 dyne = $10^{-5}$ newtons, so 1 dyne-sec/cm = $10^{-3}$ newton-sec/metre. Or stay with cgs units if you prefer, where the acceleration of gravity is $980 \hbox{cm}/\hbox{sec}^2$.

3.9.6 Solution

3.9.12 Solution

3.9.14 Solution

    

Note: to get the units right in 3.9.6, you should convert the quantities given in centimetres to metres.

E.1      A bungee jumper of mass $m$ drops from the top of a tall tower, attached to one end of an elastic cord of length $L$ whose other end is attached to the top of the tower. When the cord is stretched beyond length $L$, it obeys Hooke's Law with a spring constant $k$. We neglect the mass of the cord, and treat the jumper as a point. Note that the cord exerts no force when the distance from the jumper to the top of the tower is less than $L$.

(a)      If there is no air resistance or other damping (and no collision with the ground), how far down will the jumper get? How far up will the jumper bounce back?

(b)      For cords of different lengths made from the same material, $k$ is proportional to $1/L$. How does the maximum force exerted by the cord on the jumper depend on $L$?

(c)      Suppose there is some damping in the cord, with damping constant $\gamma$ (but still no air resistance). Assume all the constants are known. Show how you could determine whether the cord is stretched beyond length $L$ at the top of the first bounce. Don't try to do this all in one step, because the formulas will get too complicated.

Solution

E.2      The picture below shows the motion of a body hanging from a spring with no damping. The support of the spring is moved up and down as shown in the top curve, thus supplying an external force to the system. At $t=0$ the body is at rest at its equilibrium position. Note the ``beats'': the body oscillates between two sinusoidal curves which are shown in green.

(a)      Find the angular frequencies $\omega$ (for the forcing term) and $\omega_0$ (the natural angular frequency of the system) using measurements on the picture. Use two different methods to obtain $\omega_0$, and see how closely your results agree.

(b)      Is it possible to estimate $m$ or $k$ from this picture?

Solution