Chapter 1: Mechanics

A ball with mass M, moving horizontally at 4.00 m/s, collides elastically with a block with mass 3M that is initially hanging at rest from the ceiling on the end of a 50.0-cm wire. Find the maximum angle through which the block swings after it is hit?

Question:A$\mathbf{40}.\mathbf{0}-\mathbf{kg}$ packing case is initially at rest on the floor of a$\mathbf{1500}-\mathbf{kg}$ pickup truck. The coefficient of static friction between the case and the truck floor is 0.30, and the coefficient of kinetic friction is 0.20. Before each acceleration given below, the truck is traveling due north at constant speed. Find the magnitude and direction of the friction force acting on the case (a) when the truck accelerates at$\mathbf{2}.\mathbf{20}\mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$ northward and (b) when it accelerates at$\mathbf{3}.\mathbf{40}\mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$ southward.

A DNA molecule, with its double- helix structure, can in some situations behave like a spring. Measuring the force required to stretch single DNA molecules under various conditions can provide information about the biophysical properties of DNA. A technique for measuring the stretching force makes use of a very small cantilever, which consists of a beam that is supported at one end and is free to move at the other end, like a tiny diving board. The cantilever is constructed so that it obeys Hooke’s law—that is, the displacement of its free end is proportional to the force applied to it. Because different cantilevers have different force constants, the cantilever’s response must first be calibrated by applying a known force and determining the resulting deflection of the cantilever. Then one end of a DNA molecule is attached to the free end of the cantilever, and the other end of the DNA molecule is attached to a small stage that can be moved away from the cantilever, stretching the DNA. The stretched DNA pulls on the cantilever, deflecting the end of the cantilever very slightly. The measured deflection is then used to determine the force on the DNA molecule

Based on given figure below, how much elastic potential energy is stored in the DNA when it stretched\(0.1{\rm{ pN/nm}}\)?

1. \({\rm{2}}{\rm{.5}} \times {\rm{1}}{{\rm{0}}^{ - 19}}{\rm{ J}}\)
2. \(1.2 \times {\rm{1}}{{\rm{0}}^{ - 19}}{\rm{ J}}\)
3. \(5.0 \times {\rm{1}}{{\rm{0}}^{ - 19}}{\rm{ J}}\)
4. \({\rm{2}}{\rm{.5}} \times {\rm{1}}{{\rm{0}}^{ - 12}}{\rm{ J}}\)

About how long does it take a seed launched at 90° at the highest possible initial speed to reach its maximum height? Ignore air resistance. (a) 0.23 s; (b) 0.47 s; (c) 1.0 s; (d) 2.3 s.

Question: If the coefficient of static friction between a table and a uniform, massive rope is ${\mathbf{\mu }}_{\mathbf{s}}$ , what fraction of the rope can hang over the edge of the table without the rope sliding?

A liquid flowing from a vertical pipe has a definite shape as it flows from the pipe. To get the equation for this shape, assume that the liquid is in free fall once it leaves the pipe. Just as it leaves the pipe, the liquid has speed ${\mathit{v}}_{\mathbf{0}}$ and the radius of the stream of liquid is ${\mathit{r}}_{\mathbf{0}}$ . (a) Find an equation for the speed of the liquid as a function of the distance y it has fallen. Combining this with the equation of continuity, find an expression for the radius of the stream as a function of y. (b) If water flows out of a vertical pipe at a speed of 1.2 km/s , how far below the outlet will the radius be one-half the original radius of the stream?

Question: In your physics lab you release a small glider from rest at various points on a long, frictionless air track that is inclined at an angle above the horizontal. With an electronic photocell, you measure the time it takes the glider to slide a distance from the release point to the bottom of the track. Your measurements are given in Fig. P2.84, which shows a second-order polynomial (quadratic) fit to the plotted data. You are asked to find the glider’s acceleration, which is assumed to be constant. There is some error in each measurement, so instead of using a single set of x and t values, you can be more accurate if you use graphical methods and obtain your measured value of the acceleration from the graph. (a) How can you re-graph the data so that the data points fall close to a straight line? (Hint: You might want to plot or , or both, raised to some power.) (b) Construct the graph you described in part (a) and find the equation for the straight line that is the best fit to the data points. (c) Use the straight line fit from part (b) to calculate the acceleration of the glider. (d) The glider is released at a distance x = 1.35 from the bottom of the track. Use the acceleration value you obtained in part (c) to calculate the speed of the glider when it reaches the bottom of the track.

All birds, independent of their size, must maintain a power output of 10 - 25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the steady power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

Neutron Star Glitches. Occasionally, a rotating neutron star (see Exercise 10.41) undergoes a sudden and unexpected speedup called a glitch. One explanation is that a glitch occurs when the crust of the neutron star settles slightly, decreasing the moment of inertia about the rotation axis. A neutron star with angular speed${\mathbf{\omega }}_{\mathbf{0}}\mathbf{=}\mathbf{70}\mathbf{.}\mathbf{4}\mathbf{rad}\mathbf{/}\mathbf{s}$underwent such a glitch in October 1975 that increased its angular speed to $\mathbf{\omega }\mathbf{=}{\mathbf{\omega }}_{\mathbf{0}}\mathbf{+}\mathbf{∆}\mathbf{\omega }$, where$\frac{\mathbf{∆}\mathbf{\omega }}{{\mathbf{\omega }}_{\mathbf{0}}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{01}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}$. If the radius of the neutron star before the glitch was 11 km, by how much did its radius decrease in the starquake? Assume that the neutron star is a uniform sphere.

Two uniform solid spheres, each with mass$\mathbf{M}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{800}\mathbf{}\mathbf{kg}$and radius$\mathbf{R}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0800}\mathbf{}\mathbf{m}$, are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant$\mathbf{k}\mathbf{=}\mathbf{160}\mathbf{}\mathbf{N}\mathbf{/}\mathbf{m}$has one end attached to the wall and the other end attached to a frictionless ring that passes over the rod at the centre of mass of the spheres, which is midway between the centres of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the centre of mass of the spheres is simple harmonic and calculate the period.