Chapter 1: Mechanics

Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.90). (a) Which way will the system move when the blocks are released from rest? (b) What is the acceleration of the blocks? (c) What is the tension in the cord?

Figure P6.90 shows the results of measuring the force F exerted on both ends of a rubber band to stretch it a distance x from its unstretched position. (Source: www.sciencebuddies.org) The data points are well fit by the equation $\mathit{F}\mathbf{=}\mathbf{33}\mathbf{.}\mathbf{55}{\mathit{x}}^{\mathbf{0}\mathbf{.}\mathbf{4571}}$, where F is in newtons and x is in meters. (a) Does this rubber band obey Hooke’s law over the range of x shown in the graph? Explain. (b) The stiffness of a spring that obeys Hooke’s law is measured by the value of its force constant k, where k = F/x. This can be written as k = dF/dx to emphasize the quantities that are changing. Define ${\mathbf{k}}_{\mathbf{eff}}\mathbf{=}\mathbf{dF}\mathbf{/}\mathbf{dx}$ and calculate k_eff as a function of x for this rubber band. For a spring that obeys Hooke’s law, is constant, independent of x. Does the stiffness of this band, as measured by k_eff, increase or decrease as x is increased, within the range of the data? (c) How much work must be done to stretch the rubber band from x = 0 to x = 0.0400 m ? From x=0.0400 m to x=0.0800 m ? (d) One end of the rubber band is attached to a stationary vertical rod, and the band is stretched horizontally 0.0800 m from its unstretched length. A 0.300-kg object on a horizontal, frictionless surface is attached to the free end of the rubber band and released from rest. What is the speed of the object after it has traveled 0.0400 m?

You hang various masses m from the end of a vertical, 0.250-kg spring that obeys Hooke’s law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace m in the equation \(T = 2\pi \sqrt {\frac{m}{k}} \) \(m + {m_{eff}}\) where \({m_{eff}}\) is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations, obtaining these data:

(a) Graph the square of the period T versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the spring’s effective mass. (d) What fraction is \({m_{eff}}\) of the spring’s mass? (e) If a 0.450-kg mass oscillates on the end of the spring, find its period, frequency, and angular frequency

Question:Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.90). (a) Which way will the system move when the blocks are released from rest? (b) What is the acceleration of the blocks? (c) What is the tension in the cord?

The human circulatory system is closed—that is, the blood pumped out of the left ventricle of the heart into the arteries is constrained to a series of continuous, branching vessels as it passes through the capillaries and then into the veins as it returns to the heart. The blood in each of the heart’s four chambers comes briefly to rest before it is ejected by contraction of the heart muscle.

If the contraction of the left ventricle lasts 250ms and the speed of blood flow in the aorta (the large artery leaving the heart) is 0.80 m/s at the end of the contraction, what is the average acceleration of a red blood cell as it leaves the heart? (a) 310 m/s²; (b) 31 m/s²; (c) 3.2 m/s²; (d) 0.32 m/s².

The maximum force the muscles of the diaphragm can exert is 24,000 N. What maximum pressure difference can the diaphragm withstand? (a) 160 mm Hg; (b) 760 mm Hg; (c) 920 mm Hg;(d) 5000 mm Hg.

A block with mass m is revolving with linear speed ${\mathit{v}}_{\mathbf{1}}$ in a circle of radius${\mathit{r}}_{\mathbf{1}}$ on a frictionless horizontal surface (see Fig. E10.40). The string is slowly pulled from below until the radius of the circle in which the block is revolving is reduced to ${\mathit{r}}_{\mathbf{2}}$. (a) Calculate the tension T in the string as a function of r, the distance of the block from the hole. Your answer will be in terms of the initial velocity ${\mathit{v}}_{\mathbf{1}}$and the radius ${\mathit{r}}_{\mathbf{1}}$. (b) Use $\mathit{W}\mathbf{=}{\mathbf{\int }}_{{\mathbf{r}}_{\mathbf{1}}}^{{\mathbf{r}}_{\mathbf{2}}}\overrightarrow{\mathbf{T}}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{.}\mathit{d}\overrightarrow{\mathbf{r}}$to calculate the work done by $\overrightarrow{\mathbf{T}}$when r changes from ${\mathit{r}}_{\mathbf{1}}$to ${\mathit{r}}_{\mathbf{2}}$. (c) Compare the results of part (b) to the change in the kinetic energy of the block.

On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of $\mathit{v}\mathbf{=}\mathbf{1}\mathbf{.25}\mathbf{\text{\hspace{0.17em}m/s}}$. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise 9.20.) Let’s see what angular acceleration is required to keep constant. The equation of a spiral is $\mathit{r}\left(\theta \right)\mathbf{=}{\mathit{v}}_{\mathbf{0}}\mathbf{+}\mathit{\beta }\mathit{\theta }$, where ${\mathit{r}}_{\mathbf{0}}$ is the radius of the spiral at $\mathit{u}\mathbf{=}\mathbf{0}$and $\mathit{\beta }$is a constant. On a CD, ${\mathit{r}}_{\mathbf{0}}$is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, $\mathit{\beta }$must be positive so that $\mathit{r}$increases as the disc turns and $\mathit{\theta }$ increases.

1. When the disc rotates through a small angle $\mathit{d}\mathit{\theta }$, the distance scanned along the track is $\mathit{d}\mathit{s}\mathbf{=}\mathit{r}\mathit{d}\mathit{\theta }$. Using the above expression for $\mathit{r}\left(\theta \right)$, integrate $\mathit{d}\mathit{s}$to find the total distance $\mathit{s}$scanned along the track as a function of the total angle $\mathit{\theta }$through which the disc has rotated.
2. Since the track is scanned at a constant linear speed $\mathit{v}$, the distance found in part (a) is equal to. Use this to find $\mathit{\theta }$as a function of time. There will be two solutions for $\mathit{\theta }$; choose the positive one, and explain why this is the solution to choose.
3. Use your expression for $\mathit{r}\left(\theta \right)$to find the angular velocity ${\mathit{\omega }}_{\mathbf{z}}$ and the angular acceleration${\mathit{\alpha }}_{\mathbf{z}}$as functions of time. Is ${\mathit{\alpha }}_{\mathbf{z}}$constant?
4. On a CD, the inner radius of the track is $\mathbf{25}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{mm}$, the track radius increases by 1.55µm per revolution, and the playing time is 74.00 min . Find r₀, β, and the total number of revolutions made during the playing time.
5. Using your results from parts (c) and (d), make graphs of ω_z(in rad/s) versust andα_z(in rad/s²) versus t between t=0and t=74.00 min.

An angler hangs a 4.50 kgfish from a vertical steel wirelong and$\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}{\mathbf{cm}}^{\mathbf{2}}$in a cross-sectional area. The upper end of the wire is securely fastened to a support.

The angler now applies a varying forceat the lower end of the wire, pulling it very slowly downward byfrom its equilibrium position. For this downward motion, calculate

1. The work done by gravity;
2. The work done by the force$\overrightarrow{\mathbf{F}}$,
3. The work done by the force the wire exerts on the fish;
4. The change in the elastic potential energy (the potential energy associated with the tensile stress in the wire). Compare the answers in parts (d) and (e).

In terms of m₁, m₂, and g, find the acceleration of each block in Fig. P5.91. There is no friction anywhere in the system.