Spring

A smooth, thin horizontal rod OA can rotate about a vertical axis MN passing through O. Two identical small objects, each with mass $m_1 = m_2 = m$, are threaded onto the rod. They are connected by two identical light springs, each with spring constant $k$ and natural length $L_0$. The first spring connects the axis at O to mass $m_1$, and the second spring connects mass $m_1$ to mass $m_2$.

A spring with a pointer is hung next to a scale marked in millimeters. When a package of weight $W_1 = 110$ N is hung, the pointer indicates $x_1 = 40$ mm. When a package of weight $W_2 = 240$ N is hung, the pointer indicates $x_2 = 60$ mm. A third package of unknown weight $W$ is hung, and the pointer indicates $x_3 = 30$ mm.

Figure applies to the spring in a cork gun; it shows the spring force as a function of the stretch or compression of the spring. The spring is compressed by 5.5 cm and used to propel a 3.8 g cork from the gun.

A block of mass $m = 12$ kg is released from rest on a frictionless incline of angle $\theta = 30^\circ$. Below the block is a spring that can be compressed 2.0 cm by a force of 270 N. The block momentarily stops when it compresses the spring by 5.5 cm.

An 1800 kg elevator cab's cable snaps when it is at rest. The cab bottom is a distance $d = 3.7$ m above a spring with a spring constant of $k = 0.15$ MN/m. A safety device provides a constant frictional force of $f_k = 4.4$ kN that opposes the cab's motion.

A small ball of mass $m$ is connected between two light vertical elastic ropes. The other ends of the ropes are fixed at points O (above) and O' (below) on the same vertical line. The spring constants are $k_1$ for the upper rope and $k_2$ for the lower rope. When the ball is at rest at its equilibrium position C, the upper and lower ropes are stretched by lengths $l_1$ and $l_2$ respectively. The ball is then pulled vertically downward by a distance $A$ from C and released from rest.

A block P of mass $m$ is attached to a spring (constant $k$) on a smooth horizontal surface. Another block Q of mass $\beta m$ is in contact with P. A wall is at a distance $L$ from Q's initial position at equilibrium. The blocks are compressed by $L_0$ and released. After separation at the equilibrium point ($x=0$), P completes one full oscillation and then collides with Q, which has traveled to the wall and back. All collisions are perfectly elastic.