Mechanics

A $\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{\text{ kg}}$ rock is sliding on a rough, horizontal surface at $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$ and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is $\mathbf{0}\mathbf{.}\mathbf{200}$. What average power is produced by friction as the rock stops?

We want to hang a thin hoop on a horizontal nail and have the hoop make one complete small-angle oscillation each \({\bf{2}}.{\bf{0}}\,{\bf{s}}\). What must the hoop’s radius be?

A baseball thrown at an angle of  $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal strikes a building 18.0 m away at a point  8.0 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the ball’s initial velocity (the velocity with which the ball is thrown). (b) Find the magnitude and direction of the velocity of the ball just before it strikes the building.

A rocket carrying a satellite is accelerating straight up from the earth’s surface. At \({\bf{1}}.{\bf{15}}\;{\bf{s}}\) after liftoff, the rocket clears the top of its launch platform, \({\bf{63}}\;{\bf{m}}\) above the ground. After an additional \({\bf{4}}.{\bf{75}}\;{\bf{s}}\), it is \({\bf{1}}.{\bf{00}}\;{\bf{km}}\) above the ground. Calculate the magnitude of the average velocity of the rocket for (a) the \({\bf{4}}.{\bf{75}} - {\bf{s}}\) part of its flight and (b) the first \({\bf{5}}.{\bf{90}}\;{\bf{s}}\) of its flight.

A \({\bf{1}}.{\bf{80}} - {\bf{kg}}\) connecting rod from a car engine is pivoted about a horizontal knife edge as shown in Fig. E14.55. The center of gravity of the rod was located by balancing and is \({\bf{0}}.{\bf{200}}\,{\bf{m}}\) from the pivot. When the rod is set into small-amplitude oscillation, it makes \(100\) complete swings in \({\bf{120}}\,{\bf{s}}\). Calculate the moment of inertia of the rod about the rotation axis through the pivot.

The magnetic poles of a small cyclotron produce a magnetic field with magnitude \(0.85\,{\rm{T}}\). The poles have a radius of \(0.40\,{\rm{m}}\), which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons (\(q = 1.60 \times {10^{ - 19}}\,{\rm{C}}\), \(m = 1.67 \times {10^{ - 27}}\,{\rm{kg}}\)) can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For \(B = 0.85\,{\rm{T}}\), what is the maximum energy to which alpha particles (\(q = 3.20 \times {10^{ - 27}}\,{\rm{C}}\), \(m = 6.64 \times {10^{ - 27}}\,{\rm{kg}}\)) can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

Biological tissues are typically made up of $98\%$ water. Given that the density of water is $1.0\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3$, estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of $0.5\mu \mathrm{m}$; (c) a honeybee.

A circular saw blade with a radius of 0.120m starts from rest and turns in a vertical plane with a constant angular acceleration of     2.00 rev/s sq. After the blade has turned through 155 rev, a small piece of the blade breaks loose, it travels with a velocity that is initially horizontal and equal to the tangential velocity of the rim of the blade. The piece travels a vertical distance of 0.820m to the floor. How far does the piece travel horizontally, from where it broke off the blade until it strikes the floor.

The two pendulums shown in Fig. E14.57 each consist of a uniform solid ball of mass \(M\) supported by a rigid massless rod, but the ball for pendulum \(A\) is very tiny while the ball for pendulum \(B\) is much larger. Find the period of each pendulum for small displacements. Which ball takes longer to complete a swing?

A particle of charge \(q > 0\) is moving at speed \(v\) in the \( + z\)-direction through a region of uniform magnetic field \(\vec B\). The magnetic force on the particle is \(\vec F = {F_0}\left( {3\hat i + 4\hat j} \right)\), where \({F_0}\) is a positive constant. (a) Determine the components\({B_x}\), \({B_y}\), and\({B_z}\), or at least as many of the three components as is possible from the information given. (b) If it is given in addition that the magnetic field has magnitude\(6{F_0}/qv\), determine as much as you can about the remaining components of \(\vec B\).

In Canadian football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. The bar is 10.0 ft above the ground, and the ball is kicked from ground level, 36.0 ft horizontally from the bar (Fig. P3.58). Football regulations are stated in English units, but convert them to SI units for this problem. (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is kicked. What is this angle? (b) If the ball is kicked at $\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal, what must its initial speed be if it is just to clear the bar? Express your answer in m/s and in km/h .

A disk of radius $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{\text{ cm}}$ is free to turn about an axle perpendicular to it through its center. It has very thin but strong string wrapped around its rim, and the string is attached to a ball that is pulled tangentially away from the rim of the disk (Fig. P9.59). The pull increases in magnitude and produces an acceleration of the ball that obeys the equation $\mathit{a}\left(t\right)\mathbf{=}\mathit{A}\mathit{t}$, where $\mathit{t}$ is in seconds and $\mathit{A}$ is a constant. The cylinder starts from rest, and at the end of the third second, the ball’s acceleration is $\mathbf{1}\mathbf{.}\mathbf{80}\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\text{s}}^{\text{2}}}$}\right.$. (a) Find $\mathit{A}$. (b) Express the angular acceleration of the disk as a function of time. (c) How much time after the disk has begun to turn does it reach an angular speed of $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{${\displaystyle \mathbf{\text{rad}}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \mathbf{\text{s}}}$}\right.$? (d) Through what angle has the disk turned just as it reaches $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{${\displaystyle \mathbf{\text{rad}}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \mathbf{\text{s}}}$}\right.$? (Hint: See Section 2.6.)

A disk of radius $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{\text{ cm}}$ is free to turn about an axle perpendicular to it through its center. It has very thin but strong string wrapped around its rim, and the string is attached to a ball that is pulled tangentially away from the rim of the disk (Fig. P9.59). The pull increases in magnitude and produces an acceleration of the ball that obeys the equation $\mathit{a}\left(t\right)\mathbf{=}\mathit{A}\mathit{t}$, where $\mathit{t}$ is in seconds and $\mathit{A}$ is a constant. The cylinder starts from rest, and at the end of the third second, the ball’s acceleration is $\mathbf{1}\mathbf{.}\mathbf{80}\raisebox{1ex}{${\displaystyle \mathbf{\text{m}}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\mathbf{\text{s}}}^{\mathbf{2}}}$}\right.$. (a) Find $\mathit{A}$. (b) Express the angular acceleration of the disk as a function of time. (c) How much time after the disk has begun to turn does it reach an angular speed of $\mathbf{15}\mathbf{.}\mathbf{0}\raisebox{1ex}{${\displaystyle \mathbf{\text{rad}}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \mathbf{\text{s}}}$}\right.$? (d) Through what angle has the disk turned just as it reaches $\mathbf{15}\mathbf{.}\mathbf{0}\raisebox{1ex}{${\displaystyle \mathbf{\text{rad}}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \mathbf{\text{s}}}$}\right.$? (Hint: See Section 2.6.)

You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{\text{ cm}}$. Starting from rest at $\mathit{t}\mathbf{=}\mathbf{0}$, the flywheel rotates with constant angular acceleration  $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{rad}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\text{s}}^{\text{2}}}$}\right.$ about an axis perpendicular to the flywheel at its center. If the flywheel has a density (mass per unit volume) of , what thickness must it have to store $\mathbf{\text{800 J}}$ of kinetic energy at $\mathit{t}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{\text{ s}}$?

You must design a device for shooting a small marble vertically upward. The marble is in a small cup that is attached to the rim of a wheel of radius $\mathbf{0}\mathbf{.}\mathbf{260}\mathbf{\text{ m}}$; the cup is covered by a lid. The wheel starts from rest and rotates about a horizontal axis that is perpendicular to the wheel at its center. After the wheel has turned through $\mathbf{20}\mathbf{\text{ rev}}$, the cup is the same height as the center of the wheel. At this point in the motion, the lid opens and the marble travels vertically upward to a maximum height   $\mathit{h}$ above the center of the wheel. If the wheel rotates with a constant angular acceleration  , what value of $\mathit{a}$ is required for the marble to reach a height of $\mathit{h}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{\text{ m}}$?

Measurements indicate that 27.83 % of all rubidium atoms currently on the earth are the radioactive ${}^{\mathbf{87}}\mathit{R}\mathit{b}$ isotope. The rest are the stable ${}^{\mathbf{85}}\mathit{R}\mathit{b}$ isotope. The half-life of ${}^{\mathbf{87}}\mathit{R}\mathit{b}$ is $\mathbf{4}\mathbf{.}\mathbf{75}\mathbf{\times }{\mathbf{10}}^{\mathbf{10}}$ y. Assuming that no rubidium atoms have been formed since, what percentage of rubidium atoms were ${}^{\mathbf{87}}\mathit{R}\mathit{b}$ when our solar system was formed $\mathbf{4}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{9}}$y ago?

Problem 9.62: Engineers are designing a system by which a falling mass $\mathit{m}$ imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (Fig. P9.62). There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is $\mathbf{3}\mathbf{.}\mathbf{71}\raisebox{1ex}{$\mathbf{\text{m}}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{\text{s}}}^{\mathbf{2}}$}\right.$. In the earth tests, when $\mathit{m}$ is set to $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{kg}}$  and allowed to fall through $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{m}}$, it gives $\mathbf{250}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{J}}$ of kinetic energy to the drum. (a) If the system is operated on Mars, through what distance would the $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{kg}}$ mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{kg}}$ mass be moving on Mars just as the drum gained $\mathbf{250}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{J}}$ of kinetic energy?

The nucleus ${}_{\mathbf{8}}{}^{\mathbf{15}}\mathit{O}$ has a half-life of 122.2 s; ${}_{\mathbf{8}}{}^{\mathbf{19}}\mathit{O}$ has a half-life of 26.9s. If at some time a sample contains equal amounts of ${}_{\mathbf{8}}{}^{\mathbf{15}}\mathit{O}$ and ${}_{\mathbf{8}}{}^{\mathbf{19}}\mathit{O}$, what is the ratio of ${}_{\mathbf{8}}{}^{\mathbf{15}}\mathit{O}$ to ${}_{\mathbf{8}}{}^{\mathbf{19}}\mathit{O}$ (a) after 3.0 min and (b) after 12.0 min?

Two small spheres with mass m = 15.0 g are hung by silk threads of length L = 1.20 m from a common point (Fig. P21.62). When the spheres are given equal quantities of negative charge, so that q₁ = q₂ = q, each thread hangs at u = 25.0° from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of q. (c) Both threads are now shortened to length L = 0.600 m, while the charges q₁ and q₂ remain unchanged. What new angle will each thread make with the vertical? (Hint: This part of the problem can be solved numerically by using trial values for 𝛉 and adjusting the values of 𝛉 until a self-consistent answer is obtained.)

A computer disk drive is turned on starting from rest

and has constant angular acceleration. If it took 0.0865 s for the

drive to make its second complete revolution, (a) how long did it

take to make the first complete revolution, and (b) what is its

angular acceleration, in rad/s sq

In the 1986 disaster at the Chernobyl reactor in eastern Europe, about 18 of the 137Cs present in the reactor was released.

The isotope 137Cs has a half-life of 30.07 y for b decay, with the emission of a total of 1.17 MeV of energy per decay. Of this, 0.51 MeV goes to the emitted electron; the remaining 0.66 MeV goes to a g ray. The radioactive 137Cs is absorbed by plants, which are eaten by livestock and humans. How many 137Cs atoms would need to be present in each kilogram of body tissue if an equivalent dose for one week is 3.5 Sv? Assume that all of the energy from the decay is deposited in 1.0 kg of tissue and that the RBE of the electrons is 1.5.

In the human arm, the forearm and hand pivot about the elbow joint. Consider a simplified model in which the biceps muscle is attached to the forearm $\mathbf{3}\mathbf{.}\mathbf{80}\mathit{c}\mathit{m}$  from the elbow joint. Assume that the person’s hand and forearm together weigh $\mathbf{15}\mathbf{.}\mathbf{0}\mathit{N}$  and that their center of gravity is $\mathbf{15}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$  from the elbow (not quite halfway to the hand). The forearm is held horizontally at a right angle to the upper arm, with the biceps muscle exerting its force perpendicular to the forearm. (a) Draw a free-body diagram for the forearm, and find the force exerted by the biceps when the hand is empty. (b) Now the person holds an $\mathbf{80}\mathbf{.}\mathbf{0}\mathit{N}$  weight in his hand, with the forearm still horizontal. Assume that the center of gravity of this weight is $\mathbf{33}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$   from the elbow. Draw a free-body diagram for the forearm, and find the force now exerted by the biceps. Explain why the biceps muscle needs to be very strong. (c) Under the conditions of part (b), find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) While holding the  $\mathbf{80}\mathbf{.}\mathbf{0}\mathit{N}$ weight, the person raises his forearm until it is at an angle of  $\mathbf{53}\mathbf{.0}\mathbf{\text{°}}$ above the horizontal. If the biceps muscle continues to exert its force perpendicular to the forearm, what is this force now? Has the force increased or decreased from its value in part (b)? Explain why this is so, and test your answer by doing this with your own arm.

A small block with mass \(0.0400\,{\rm{kg}}\) slides in a vertical circle of radius \(R = 0.500\,{\rm{m}}\) on the inside of a circular track. During one of the revolution of the block, when the block is the bottom of its path, point A, the normal force extended on the block by the track has magnitude \(3.95\,{\rm{N}}\). In this same revolution, when the block reaches the top of its path, point B, the normal force exerted on the block has magnitude \(0.680\,{\rm{N}}\). How much work is done on the block by friction during the motion of the block from point A to point B?

Question: A solid uniform ball rolls without slipping up a hill (Fig. P10.70). At the top of the hill, it is moving horizontally, and then it goes over the vertical cliff. (a) How far from the foot of the cliff does the ball land, and how fast is it moving just before it lands? (b) Notice that when the balls lands, it has a greater translational speed than when it was at the bottom of the hill. Does this mean that the ball somehow gained energy? Explain!

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: \[\left( \mathbf{a} \right)\underline{\phantom{xxx}}\mathbf{uus},\underline{\phantom{xxx}}\left( \mathbf{b} \right)\underline{\phantom{xxx}}\mathbf{c\bar{s}},\underline{\phantom{xxx}}\left( \mathbf{c} \right)\underline{\phantom{xxx}}\mathbf{\bar{d}\bar{d}\bar{u}},\] and \[\left( \mathbf{d} \right)\underline{\phantom{xxx}}\mathbf{\bar{c}b}.\]

Two boxes connected by a light horizontal rope are on a horizontal surface (Fig. E5.37). The coefficient of kinetic friction between each box and the surface is ${\mathbf{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{30}$.Box B has mass 5.00 kg , and box has mass . A force F with magnitude 40.0 N and direction $\mathbf{53}\mathbf{.}\mathbf{1}\mathbf{°}$ above the horizontal is applied to the box, and both boxes move to the right with $\mathbf{a}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. (a) What is the tension in the rope that connects the boxes? (b) What is ?

Radioisotopes are used in a variety of manufacturing and testing techniques. Wear measurements can be made using the following method. An automobile engine is produced using piston rings with a total mass of 100 g, which includes 9.4 m Ci of whose half-life is 45 days. The engine is test-run for 1000 hours, after which the oil is drained and its activity is measured. If the activity of the engine oil is 84 decays/s, how much mass was worn from the piston rings per hour of operation?

Question: You are designing a system for moving aluminum cylinders from the ground to a loading dock. You use a sturdy wooden ramp that is 6.00 m long and inclined at \(37^\circ \) above the horizontal. Each cylinder is fitted with a light, frictionless yoke through its center, and a light (but strong) rope is attached to the yoke. Each cylinder is uniform and has mass 460 kg and radius 0.300 m. The cylinders are pulled up the ramp by applying a constant force \(\vec F\) to the free end of the rope. \(\vec F\) is parallel to the surface of the ramp and exerts no torque on the cylinder. The coefficient of static friction between the ramp surface and the cylinder is 0.120. (a) What is the largest magnitude \(\vec F\) can have so that the cylinder still rolls without slipping as it moves up the ramp? (b) If the cylinder starts from rest at the bottom of the ramp and rolls without slipping as it moves up the ramp, what is the shortest time it can take the cylinder to reach the top of the ramp?

Question: A 42.0-cm-diameter wheel, consisting of a rim and six spokes, is constructed from a thin, rigid plastic material having a linear mass density of \(25.0\;{\rm{g/cm}}\). This wheel is released from rest at the top of a hill 58.0 m high. (a) How fast is it rolling when it reaches the bottom of the hill? (b) How would your answer change if the linear mass density and the diameter of the wheel were each doubled?

A uniform solid cylinder with mass M and radius 2R rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string (Fig. P10.75). The string doesn’t slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

In the methane molecule, $\mathit{C}{\mathit{H}}_{\mathbf{4}}$ , each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the $\mathit{C}\mathbf{-}\mathit{H}$  bonds is in the direction of $\hat{i} -\hat{j}-\hat{\kappa }$ , an adjacent $\mathit{C}\mathbf{ }\mathbf{-}\mathbf{ }\mathit{H}$bond is in the $\hat{i}-\hat{j}-\hat{\kappa }$ direction. Calculate the angle between these two bonds.

Question: The solid wood door of a gymnasium is 1.00 m wide and 2.00 m high, has total mass 35.0 kg, and is hinged along one side. The door is open and at rest when a stray basketball hits the center of the door head-on, applying an average force of 1500 N to the door for 8.00 ms. Find the angular speed of the door after the impact. (Hint: Integrating Eq. (10.29) yields \(\Delta {L_Z} = \int_{t1}^{t2} {\left( {\sum {\tau _z}} \right)} dt = {\left( {\sum {\tau _z}} \right)_{av}}\Delta t.\)The quantity \(\int_{t1}^{t2} {\left( {\sum {\tau _z}} \right)} dt\) is called the angular impulse.)

Vectors $\overrightarrow{A}$  and $\overrightarrow{B}$  have scalar product $-6.00$, and their vector product has magnitude $+9.00$ . What is the angle between these two vectors?

A 40.0-N force stretches a vertical spring 0.250 m. (a) What mass must be suspended from the spring so that the system will oscillate with a period of 1.00 s? (b) If the amplitude of the motion is 0.050 m and the period is that specified in part (a), where is the object and in what direction is it moving 0.35 s after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is 0.030 m below the equilibrium position, moving upward?

An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. [Note: The gravitational force on the object as a function of the object’s distance r from the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration \({a_x}\) and the displacement from equilibrium x are related by Eq. (14.8), and the period is then\(T = \frac{{2\pi }}{\omega }\)].

A 40.0 - kg packing case is initially at rest on the floor of a 1500- 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 2.20$\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ northward and (b) when it accelerates at $\mathbf{3}\mathbf{.}\mathbf{40}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ southward.

A uniformly charged disk like the disk in Fig. 21.25 has a radius of 2.50 cm and carries a total charge of 7.0 * 10-12 C. 

(a) Find the electric field (magnitude and direction) on the x-axis at x = 20.0 cm

. 

(b) Show that for x W R, becomes E = Q>4pP0x2, where Q is the total charge on the disk. 

(c) Is the magnitude of the electric field you calculated in part (a) larger or smaller than the electric field 20.0 cm from a point charge that has the same total charge as this disk? In terms of the approximation used in part (b) to derive E = Q>4pP0x2 for a point charge from Eq. explain why this is so. 

(d) What is the percent difference between the electric fields produced by the finite disk and by a point charge with the same charge at x = 20.0 cm and x = 10.0 cm?

 At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye’s T3 law: 

                                                                                       $\mathbf{C}\mathbf{=}\mathbf{K}\frac{{\mathbf{T}}^{\mathbf{3}}}{{\mathbf{\theta }}^{\mathbf{3}}}$

where $\mathbf{k}\mathbf{=}\mathbf{ }\mathbf{1940}\mathbf{ }\mathbf{J}\mathbf{/}\mathbf{mol}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{K}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{\theta }\mathbf{=}\mathbf{281}\mathbf{K}$. (a) How much heat is required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K? (Hint: Use Eq. (17.18) in the form dQ = nCdT and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at 40.0 K?

Question: The Silently Ringing Bell. A large, 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell’s center of mass is 0.60 m below the pivot. The bell’s moment of inertia about an axis at the pivot is \({\bf{18}}.{\bf{0}}{\rm{ }}{\bf{kg}} \cdot {{\bf{m}}^2}\). The clapper is a small, 1.8-kg mass attached to one end of a slender rod of length L and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length L of the clapper rod for the bell to ring silently—that is, for the period of oscillation for the bell to equal that of the clapper?

A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. If the rod is displaced by a small angle \(\theta \) from the vertical (Fig. P14.87) and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle \(\theta \) is small enough for the approximations \(\sin \;\theta   \approx \theta \;{\rm{and}}\;\cos \;\theta  \approx 1\) to be valid. The motion is simple harmonic if \({d^2}\theta /d{t^2} =  - {\omega ^2}\theta \) and the period is then \(T = 2\pi /\omega \).)

An object has several forces acting on it. One of these forces is $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{F}}\mathbf{=}\mathbf{\alpha xy}\hat{\mathbf{i}}$, a force in the x-direction whose magnitude depends on the position of the object, with $\mathbf{\alpha }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{ }\mathbf{N}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$. Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point $\left(x=0, y=3.00 m\right)$ and moves parallel to the x-axis to the point $(x=2.00, y=3.00 m)$. (b) The object starts at the point $\left(x=2.00, y=0\right)$ and moves in the y-direction to the point, $\left(x=2.00, y=3.00 m\right)$. (c) The object starts at the origin and moves on the line $\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{x}$ to the point $\left(x=2.00, y=3.00 m\right)$.

Two identical thin rods, each with mass m and length L, are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge (Fig. P14.88). If the L-shaped object is deflected slightly, it oscillates. Find the frequency of oscillation.

A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure P14.89 shows a graph of its velocity component \({v_x}\) as a function of time t. Find (a) the period, (b) the frequency, and (c) the angular frequency of this motion. (d) What is the amplitude (in cm), and at what times does the mass reach this position? (e) Find the maximum acceleration magnitude of the mass and the times at which it occurs. (f) What is the value of 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

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.

Question: When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding; a silver dollar will roll on its edge much farther than it will slide on its flat side (see Section 5.3). When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that \({a_x}\)  and \({a_z}\) are approximately zero and \({v_x}\) and\({\omega _z}\) are approximately constant. Rolling without slipping means\({v_x} = r{\omega _z}\;{\rm{and}}\;{a_x} = r{\alpha _z}\). If an object is set in motion on a surface without these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established. A solid cylinder with mass M and radius R, rotating with angular speed \({\omega _0}\) about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is \({\mu _k}\). (a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force on the cylinder. Calculate the accelerations \({a_x}\) of the center of mass and \({a_z}\) of rotation about the center of mass. (b) The cylinder is initially slipping completely, so initially \({\omega _z} = {\omega _0}\) but \({v_x} = 0\). Rolling without slipping sets in when \({v_x} = r{\omega _z}\). Calculate the distance the cylinder rolls before slipping stops. (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.

The Effective Force Constant of Two Springs. Two springs with the same unstretched length but different force constants \({k_1}\)  and \({k_2}\) are attached to a block with mass m on a level, frictionless surface. Calculate the effective force constant \({k_{eff}}\) in each of the three cases (a), (b), and (c) depicted in Fig. P14.92. (The effective force constant is defined by \(\sum {F_x} =  - {k_{eff}}x\) ) (d) An object with mass m, suspended from a uniform spring with a force constant k, vibrates with a frequency \({f_1}\). When the spring is cut in half and the same object is suspended from one of the halves, the frequency is \({f_2}\). What is the ratio \({f_1}/{f_2}\)?

A Spring with Mass. The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring’s mass, consider a spring with mass M, equilibrium length \({L_0}\), and spring constant k. When stretched or compressed to a length L, the potential energy is \(\frac{1}{2}k{x^2}\), where \(x = L - {L_0}\). 

(a) Consider a spring, as described above that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of  l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not \(\frac{1}{2}M{v^2}\), since not all of the spring moves with the same speed.) (b) Take the time derivative of the conservation of energy equation, Eq. (14.21), for a mass m moving on the end of a massless spring. By comparing your results to Eq. (14.8), which defines \(\omega \), show that the angular frequency of oscillation is \(\omega  = \sqrt {\frac{k}{m}} \). (c) Apply the procedure of part (b) to obtain the angular frequency of oscillation \(\omega \) of the spring considered in part (a). If the effective mass M′ of the spring is defined by \(\omega   = \sqrt {\frac{k}{{M'}}} \), what is M′ in terms of M?

BIO “Seeing” Surfaces at the nanoscale. One technique for making images of surfaces at the nanometer scale, including membranes and biomolecules, is dynamic atomic force microscopy. In this technique, a small tip is attached to a cantilever, which is a flexible, rectangular slab supported at one end, like a diving board. The cantilever vibrates, so the tip moves up and down in simple harmonic motion. In one operating mode, the resonant frequency for a cantilever with force constant k = 1000 N/m is 100 kHz. As the oscillating tip is brought within a few nanometers of the surface of a sample (as shown in the figure), it experiences an attractive force from the surface. For an oscillation with a small amplitude (typically, 0.050 nm), the force F that the sample surface exerts on the tip varies linearly with the displacement x of the tip, \(\left| F \right| = {k_{{\rm{surf}}}}x\), where \({k_{{\rm{surf}}}}\) is the effective force constant for this force. The net force on the tip is therefore\(\left( {k + {k_{{\rm{surf}}}}} \right)x\), and the frequency of the oscillation changes slightly due to the interaction with the surface. Measurements of the frequency as the tip moves over different parts of the sample’s surface can provide information about the sample.

If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? (a) 25 ng; (b) 100 ng; (c)\({\bf{2}}.{\bf{5}}{\rm{ }}\mu {\bf{g}}\); (d) \(100\;\mu {\bf{g}}\).