Mechanics

Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N. For each rod, what are (a) the strain and (b) the elongation?

A metal rod that is 4.00 m long and 0.50 cm² in cross-sectional area is found to stretch 0.20 cm under a tension of 5000 N. What is Young’s modulus for this metal?

A nylon rope used by mountaineers elongates 1.10m under the weight of a  65.0-kg climber. If the rope is 45.0 m in length and 7.0 mm in diameter, what is Young’s modulus for nylon?

In constructing a large mobile, an artist hangs an aluminum sphere of mass 6.0 kg from a vertical steel wire 0.50 m long and 2.5 ^* 10^-3 cm² in cross-sectional area. On the bottom of the sphere he attaches a similar steel wire, from which he hangs a brass cube of mass 10.0kg. For each wire, compute (a) the tensile strain and (b) the elongation.

A vertical, solid steel post 25 cm in diameter and 2.50 m long is required to support a load of 8000 Kg. You can ignore the weight of the post. What are (a) the stress in the post; (b) the strain in the post; and (c) the change in the post’s length when the load is applied?

The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by 0.10% of their original volume? (b) Given that the pressure in the ocean increases by 1.0 ^* 10⁴Pa for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by 0.10% ? Does it seem that bone compression is a problem she needs to be concerned with when diving?

In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 km and the pressure is 1.16^*10⁸Pa (about 1.15^*10³atm) If a cubic meter of water is taken from the surface to this depth, what is the change in its volume? (Normal atmospheric pressure is about 1.0^*10⁵Pa. Assume that   for seawater is the same as the freshwater value given in Table 11.2.) (b) What is the density of seawater at this depth? (At the surface, seawater has a density of 1.03^*10³kg/m³.

A copper cube measures 6.00 cm on each side. The bottom face is held in place by very strong glue to a flat horizontal surface, while a horizontal force F is applied to the upper face parallel to one of the edges. (Consult Table 11.1.) (a) Show that the glue exerts a force F on the bottom face that is equal in magnitude but opposite to the force on the top face. (b) How large must F be to cause the cube to deform by 0.250 mm? (c) If the same experiment were performed on a lead cube of the same size as the copper one, by what distance would it deform for the same force as in part (b)?

A square steel plate is 10.0 cm on a side and 0.500 cm thick. (a) Find the shear strain that results if a force of magnitude $\mathbf{9}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}$ N is applied to each of the four sides, parallel to the side. (b) Find the displacement x in centimeters.

In lab tests on a 9.25-cm cube of a certain material, a force of 1375 N directed at 8.50^o to the cube causes the cube to deform through an angle of 1.24^o. What is the shear modulus of material?

A brass wire is to withstand a tensile force of 350 N without breaking. What minimum diameter must the wire have?

In a materials testing laboratory, a metal wire made from a new alloy is found to break when a tensile force of 90.8 N is applied perpendicular to each end. If the diameter of the wire is 1.84 mm, what is the breaking stress of the alloy?

A 4.0-m-long steel wire has a cross-sectional area of 0.050 cm². Its proportional limit has a value of 0.0016 times its Young’s Modulus. Its breaking stress has a value of 0.0065 times its Young’s modulus. The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit? (b) How much will the wire stretch under this load? (c) What is the maximum weight that the wire can support?

A steel cable with cross-sectional area 3.00 cm² has an elastic limit of $\mathbf{2}\mathbf{.}\mathbf{40}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$ Pa. Find the maximum upward acceleration that can be given a 1200-kg elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

A door 1.00 m wide and 2.00 m high weighs 330 N and is supported by two hinges, one 0.50 m from the top and the other 0.50 m from the bottom. Each hinge supports half the total weight of the door. Assuming that the door’s center of gravity is at its center, find the horizontal components of force exerted on the door by each hinge.

A box of negligible mass rests at the left end of a 2.00-m, 25.0-kg plank. The width of the box is 75.0 cm, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is 50.0 cm from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint?

Mountaineers often use a rope to lower themselves down r = the face of a cliff (this is called rappelling). They do this with their body nearly horizontal and their feet pushing against the cliff. Suppose that an 82-0-kg climber, who is 1.90 m tall and has a center of gravity 1.1 mm from his feet, rappels down a vertical cliff with his body raised 35.0^o above the horizontal. He holds the rope 1.40 m from his feet, and it makes a 25.0^o angle with the cliff face. (a) What tension does his rope need to support? (b) Find the horizontal and vertical components of the force that the cliff face exerts on the climber’s feet. (c) What minimum coefficient of static friction is needed to prevent the climber’s feet from slipping on the cliff face if he has one foot at a time against the cliff?

A uniform, 8.0-m, 1150-kg beam is hinged to a wall and supported by a thin cable attached 2.0 m from the free end of the beam. The beam is supported at an angle of 30.0^oabove the horizontal. (a) Draw a free-body diagram of the beam. (b) Find the tension in the cable. (c) How hard does the beam push inward on the wall?

A uniform, 255-N rod that is 2.00 m long carries a 225-N weight at its right end and an unknown weight W toward the left end. When W is placed 50.0 cm from the left end of the rod, the system just balances horizontally when the fulcrum is located 75.0 cm from the right end. (a) Find W. (b) If W is now moved 25.0 cm to the right, how far and in what direction must the fulcrum be moved to restore balance?

A claw hammer is used to pull a nail out of a board. The nail is at an angle of 60^o to the board, and a force ${\overrightarrow{\mathbf{F}}}_{\mathbf{1}}$ of magnitude 400 N applied to the nail is required to pull it from the board. The hammerhead contacts the board at a point A, which is 0.800 m from where the nail enters the board. A horizontal force ${\overrightarrow{\mathbf{F}}}_{\mathbf{2}}$ is applied to the hammer handle at a distance of 0.300 m above the board. What magnitude of force ${\overrightarrow{\mathbf{F}}}_{\mathbf{2}}$ is required to apply the required force (${\overrightarrow{\mathbf{F}}}_{\mathbf{1}}$) to the nail? (Ignore the weight of the hammer.)

You open a restaurant and hope to entice customers by hanging out a sign. The uniform horizontal beam supporting the sign is 1.50 m long, has a mass of 16.0 kg, and is hinged to the wall. The sign itself is uniform with a mass of 28.0 kg and overall length of 1.20 m. The two wires supporting the sign are each 32.0 cm long, are 90.0 cm apart, and are equally spaced from the middle of the sign. The cable supporting the beam is 2.00 m long.

End A of the bar AB in the fig rests on a frictionless horizontal surface, and end B is hinged. A horizontal force $\overrightarrow{\mathbf{F}}$of magnitude 220 N is exerted on end A. Ignore the weight of the bar. What are the horizontal and vertical components of the force exerted by the bar on the hinge at B?

A therapists tells a 74-kg patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg-cast system. To comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper leg (thighs) together typically account for 21.5% of body weight and the center of mass of each thigh is 18.0 cm from the hip joint. The patient also reads that the two lower legs (including the feet) are 14.0% of body weight, with a center of mass 69.0 cm from the hip joint. The cast has a mass of 5.50 kg, and its center of mass is 78.0 cm from the hip joint.  How far from the hip joint should the supporting strap be attached to the cast?

A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, set the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to the end opposite the hinge. The drawbridge is 40.0 m long and has a mass of 18000kg; its center of gravity is at its midpoint. The cement mixer, with driver, has mass 30000 kg. When the drawbridge has an angle of 30^o above horizontal, the cable makes an angle of 70^o with the surface of the bridge.

Question: A refrigerator has a coefficient of performance of \(2.25\), runs on an input of \(135\;W\)of electrical power, and keeps its inside compartment at \(5^\circ C\). If you put a dozen \(1.0 - L\) plastic bottles of water at \(31^\circ C\) into this refrigerator, how long will it take for them to be cooled down to \(5^\circ C\)? (Ignore any heat that leaves the plastic.)

A crate with mass 32.5 kg initially at rest on a warehouse floor is acted on by a net horizontal force of 14.0 N. (a) What acceleration is produced? (b) How far does the crate travel in 10.0 s? (c) What is its speed at the end of 10.0 s?

You are preparing some apparatus for a visit to a newly discovered planet Caasi having oceans of glycerin and a surface acceleration due to gravity of 5.40 m/s2 . If your apparatus floats in the oceans on earth with 25.0% of its volume submerged, what percentage will be submerged in the glycerin oceans of Caasi?

Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere 2.00 m to the left, it nearly touches a vertical wall; with the string taut, you release the sphere from rest. The sphere swings back and forth as a simple pendulum, and you measure its period T. You repeat this act for strings of various lengths L, each time starting the motion with the sphere displaced 2.00 m to the left of the vertical position of the string. In each case the sphere’s radius is very small compared with L. Your results are given in the table:

(a) For the five largest values of L, graph \({T^2}\) versus L. Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected? (b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as L decreases. To see this effect more clearly, plot \({T \mathord{\left/ {\vphantom {T {{T_0}}}} \right.} {{T_0}}}\) versus L, where \({T_0} = 2\pi \sqrt {{L \mathord{\left/ {\vphantom {L g}} \right.} g}} \) and \(g = 9.8\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}\). (c) Use your graph of \({T \mathord{\left/ {\vphantom {T {{T_0}}}} \right.} {{T_0}}}\) versus L to estimate the angular amplitude of the pendulum (in degrees) for which the equation \(T = 2\pi \sqrt {{L \mathord{\left/ {\vphantom {L g}} \right. } g}} \) is in error by 5%.

Question: Identify the correspondence principle.

Eavesdropping! You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 db. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

Question: An electron at point A in Fig. E27.15 has a speed \({v_0}\) of 1 41 10 m/s. Find (a) the magnitude and direction of the magnetic field that will cause the electron to follow the semicircular path from A to B, and 

(b) the time required for the electron to move from A to B.

Repeat the exercise 27.15 for the case in which the particle is a proton rather than an electron.

At what point in an elliptical orbit is the acceleration maximum? At what point is it minimum? Justify your answers.

Two identical balls, A and B, are each attached to very light string, and each string is wrapped around the rim of a frictionless pulley of mass M. The only difference is that the pulley for ball A is a solid disk, while the one for ball B is a hollow disk, like part (e) in Table 9.2. If both balls are released from rest and fall the same distance, which one will have more kinetic energy, or will they have the same kinetic energy? Explain your reasoning.

Question: A 9.00-m-long uniform beam is hinged to a vertical wall and held horizontally by a 5.00-m-long cable attached to the wall 4.00 m above the hinge (Fig. E11.17). The metal of this cable has a test strength of 1.00 kN, which means that it will break if the tension in it exceeds that amount. (a) Draw a free-body diagram of the beam. (b) What is the heaviest beam that the cable can support in this configuration? (c) Find the horizontal and vertical components of the force the hinge exerts on the beam. Is the vertical component upward or downward?

Let A represent any nonzero vector. Why is $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$ a unit vector, and what is its direction? If $\theta$ is the angle that A makes with the +x-axis, explain why $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$  is called the direction cosine for that axis.

Question: A 550-N physics student stands on a bathroom scale in an elevator that is supported by a cable. The combined mass of student plus elevator is 850 kg. As the elevator starts moving, the scale reads 450 N. 

(a) Find the acceleration of the elevator (magnitude and direction). 

(b) What is the acceleration if the scale reads 670 N? 

(c) If the scale reads zero, should the student worry? Explain. 

(d) What is the tension in the cable in part (a) and (c)?

Question: An electron is trapped in a quantum dot, in which it is continued to a very small region in all three dimensions, If the lowest energy transition is to produce a photon of $\mathbf{450}\mathbf{\text{ nm}}$ wavelength, what should be the width of the well (assumed cubic)?

In Wagner’s opera Das Rheingold, the goddess Freia is ransomed for a pile of gold just tall enough and wide enough to hide her from sight. Estimate the monetary value of this pile. The density of gold is$\mathbf{19}\mathbf{.}\mathbf{3}\mathbf{ }\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ 19.3 g / cm², and take its value to be about $\mathbf{\$}\mathbf{10}$ per gram.

For a satellite to be in a circular orbit 890 km above the surface of the earth, (a) what orbital speed must it be given, and (b) what is the period of the orbit (in hours)?

How many times does a human heart beat during a person’s lifetime? How many gallons of blood does it pump? (Estimate that the heart pumps $\mathbf{50}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{3}}$ of blood with each beat.) 

Question: A 2540 kg test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force such that its vertical velocity as a function of time is given by $\mathit{v}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{A}\mathit{t}\mathbf{+}\mathit{B}{\mathit{t}}^{\mathbf{2}}$, where A and B are constants and time is measured from the instant the fuel is ignited. The rocket has an upward acceleration of $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{ }{\mathbf{\text{m/s}}}^{\mathbf{\text{2}}}$ at the instant of ignition and,1.00 s later, an upward velocity of $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{m/s}}$. (a) Determine A and B, including their SI units. (b) At 4.00 s after fuel ignition, what is the acceleration of the rocket, and (c) what thrust force does the burning fuel exert on it, assuming no air resistance? Express the thrust in newton and as a multiple of the rocket’s weight. (d) What was the initial thrust due to the fuel?

You are using water to dilute small amounts of chemicals in the laboratory, drop by drop. How many drops of water are in a $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{-}\mathit{L}$ bottle? (Hint: Start by estimating the diameter of a drop of water.)

Question: A 2.00-kg box is moving to the right with speed on a horizontal, frictionless surface. At t = 0 a horizontal force is applied to the box. The force is directed to the left and has magnitude $\mathbf{F}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{N}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{)}\mathbf{t}$. (a) What distance does the box move from its position at t = 0 before its speed is reduced to zero? (b) If the force continues to be applied, what is the speed of the box at t = 3.00s?

A relaxed biceps muscle requires a force of 25.0 N for an elongation of 3.0 cm; the same muscle under maximum tension requires a force of 500 N- for the same elongation. Find Young’s modulus for the muscle tissue under each of these conditions if the muscle is assumed to be a uniform cylinder with length 0.200 m and cross-sectional area $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{2}}$.

A rectangular loop with dimensions 4.20 cm by 9.50 cm carries current . The current in the loop produces a magnetic field at the center of the loop that has magnitude 5.50$\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{T}$ and direction away from you as you view the plane of the loop. What are the magnitude and direction (clockwise or counter clockwise) of the current in the loop?

A circular steel wire 2.00 m long must stretch no more than 0.25 cm  when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

A postal employee drives a delivery truck along the route shown in Fig. E1.25. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exercise 1.32 for a different approach.)

A jet plane at takeoff can produce sound of intensity \[10.0 {W \mathord{\left/

 {\vphantom {W {{m^2}}}} \right.

 \nulldelimiterspace} {{m^2}}}\]at \[30.0 m\]away. But you prefer the tranquil sound of normal conversation, which is\[1.0 {{\mu W} \mathord{\left/

 {\vphantom {{\mu W} {{m^2}}}} \right.

 \nulldelimiterspace} {{m^2}}}\] . Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N . For each rod, what are (a) the strain and (b) the elongation?

Compute the x- and y-components of the vectors $\stackrel{\rightharpoonup }{A}$,$\stackrel{\rightharpoonup }{B}$, $\stackrel{\rightharpoonup }{C}$ and $\stackrel{\rightharpoonup }{D}$ in Fig. E1.24.