Chapter 5

A block of mass \(M\) slides along a horizontal table with speed \(v_{0}\). At \(x=0\) it hits a spring with spring constant \(k\) and begins to experience a friction force, as indicated in the right-hand sketch. The coefficient of friction is variable and is given by \(\mu=b x\), where \(b\) is a constant. Find the distance \(l\) the block travels before coming to rest.

A small cube of mass \(m\) slides down a circular path of radius \(R\) cut into a large block of mass \(M\), as shown. \(M\) rests on a table, and both blocks move without friction. The blocks are initially at rest, and \(m\) starts from the top of the path. Find the velocity \(v\) of the cube as it leaves the block.

Mass \(m\) whirls on a frictionless table, held to circular motion by a string which passes through a hole in the table. The string is pulled so that the radius of the circle changes from \(r_{i}\) to \(r_{f}\). (a) Show that the quantity \(L=m r^{2} \dot{\theta}\) remains constant. (b) Show that the work in pulling the string equals the increase in kinetic energy of the mass.

A small block slides from rest from the top of a frictionless sphere of radius \(R\), as shown on the next page. How far below the top \(x\) does it lose contact with the sphere? The sphere does not move.

A block of mass \(M\) on a horizontal frictionless table is connected to a spring (spring constant \(k\) ). The block is set in motion so that it oscillates about its equilibrium point with a certain amplitude \(A_{0}\). The period of motion is \(T_{0}=2 \pi \sqrt{M / k}\). (a) A lump of sticky putty of mass \(m\) is dropped onto the block. The putty sticks without bouncing. The putty hits \(M\) at the instant when the velocity of \(M\) is zero. Find (1) The new period. (2) The new amplitude. (3) The change in the mechanical energy of the system. (b) Repeat part ( \(a\) ), but this time assume that the sticky putty hits \(M\) at the instant when \(M\) has its maximum velocity.

A chain of total mass \(M\) and length \(l\) is suspended vertically with its lowest end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length of chain, \(x\), has fallen? (Neglect the size of individual links.)

It is told that during World War II the Russians, lacking sufficient parachutes for airborne operations, occasionally dropped soldiers inside bales of hay onto snow. The human body can survive an average pressure on impact of \(30 \mathrm{lb} / \mathrm{in}^{2}\). Suppose that the lead plane drops a dummy bale equal in weight to a loaded one from an altitude of \(100 \mathrm{ft}\), and that the pilot observes that it sinks about \(2 \mathrm{ft}\) into the snow. If the weight of an average soldier is \(180 \mathrm{lb}\) and his effective area is \(5 \mathrm{ft}^{2}\), is it safe to drop the men?

A commonly used potential energy function to describe the interaction between two atoms is the Lennard-Jones \(6-12\) potential given by $$ U=\epsilon\left[\left(\frac{r_{0}}{r}\right)^{12}-2\left(\frac{r_{0}}{r}\right)^{6}\right] $$ (a) Find the position of the potential minimum and its value. (b) Near the minimum the atoms execute simple harmonic motion. Find the frequency of oscillation.

A particle of mass \(m\) moves in one dimension along the positive \(x\) axis. It is acted on by a constant force directed toward the origin with magnitude \(B\), and an inverse-square law repulsive force with magnitude \(A / x^{2}\) ( \(a\) ) Find the potential energy function \(U(x)\). (b) Sketch the energy diagram for the system when the maximum kinetic energy is \(K_{0}=\frac{1}{2} m v_{0}^{2}\). (c) Find the equilibrium position, \(x_{0}\).

A 1800 -lb sportscar accelerates to \(60 \mathrm{mi} / \mathrm{h}\) in \(4 \mathrm{~s}\). What is the average power that the engine delivers to the car's motion during this period? For consistency, we are using the definition \(1 \mathrm{hp}=746 \mathrm{~W}\).

A \(55-\mathrm{kg}\) athlete leaps into the air from a crouching position. Her center of mass rises \(60 \mathrm{~cm}\) as her feet leave the ground and then it continues another \(80 \mathrm{~cm}\) to the top of the leap. What is the average power she develops, assuming the force on the ground is constant?

Sand runs from a hopper at constant rate \(d m / d t\) onto a horizontal conveyor belt driven at constant speed \(V\) by a motor. ( \(a\) ) Find the power needed to drive the belt. (b) Compare the answer to \((a)\) with the rate of change of kinetic energy of the sand. Can you account for the difference?

A uniform rope of mass density \(\lambda\) per unit length is coiled on a smooth horizontal table. One end is pulled straight up with constant speed \(v_{0}\), as shown. (a) Find the force exerted on the end of the rope as a function of height \(y\). (b) Compare the power delivered to the rope with the rate of change of the rope's total mechanical energy.