Mechanics

Explain why it would not make sense to use a full-size glass thermometer to measure the temperature of a thimbleful of hot water.

15.1. The speed of sound in air at $\mathbf{20}\mathbf{°}\mathbf{C}$  is 344m/s . (a) What is the wavelength of a sound wave with a frequency of 784 Hz , corresponding to the note  5G on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher (twice the frequency) than the note impart (a)?

A 20.0L tank contains  of helium at . The molar mass of helium is 4.00 g/mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

You push your physics book $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{m}$ along a horizontal tabletop with a horizontal push of $\mathbf{2}\mathbf{.}\mathbf{4}\mathbf{ }\mathbf{N}$ while the opposing force of friction is $\mathbf{0}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{N}$. How much work does each of the following forces do on the book:

(a) your $\mathbf{2}\mathbf{.}\mathbf{4}\mathbf{ }\mathbf{N}$ push,

 (b) the friction force,

(c) the normal force from the tabletop, and 

(d) gravity?

(e) What is the net work done on the book?

Suppose you are asked to compute the tangent of 5.00meters. Is this possible? Why or why not?

 If you heat the air inside a rigid, sealed container until its Kelvin temperature doubles, the air pressure in the container will also double. Is the same thing true if you double the Celsius temperature of the air in the container? Explain.

The center of gravity of a $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{-}\mathit{k}\mathit{g}$ irregular object is shown in Fig. E11.2. You need to move the center of gravity $\mathbf{2}\mathbf{.}\mathbf{20}\mathbf{ }\mathit{c}\mathit{m}$to the left by gluing on a $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{-}\mathit{k}\mathit{g}$mass, which will then be considered as part of the object. Where should the center of gravity of this additional mass be located?

If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced $\mathbf{0}\mathbf{.}\mathbf{120}\mathbf{ }\mathit{m}$  from its equilibrium position and released with zero initial speed, then after $\mathbf{0}\mathbf{.}\mathbf{800}\mathbf{ }\mathit{s}$ its displacement is found to be $\mathbf{0}\mathbf{.}\mathbf{120}\mathbf{ }\mathit{m}$ on the opposite side, and it has passed the equilibrium position once during this interval. Find (a) the amplitude, (b) the period; (c) the frequency.

A cat walks in a straight line, which we shall call the \(x\)-axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat’s motion and construct a graph of the feline’s velocity as a function of time (Fig. E2.30). (a) Find the cat’s velocity at \(t = 4.0\,s\) and at \(t = 7.0\,s\).

The graph in Fig. E2.31 shows the velocity of a motorcycle police officer plotted as a function of time. (a) Find the instantaneous acceleration at \(t = 3\,s,\,\,t = 7\,s{\rm{  and   }}t = 11\,s\).

How many nanoseconds does it take light to travel 1.00 ft in vacuum?

Question: For an electron, apply the relationship between the de Broglie wavelength and the kinetic energy.

An object is released from rest at the top of a ramp. If the ramp is frictionless, does the object’s speed at the bottom of the ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is not frictionless?

The tip of a tuning fork goes through $\mathbf{440}$ complete vibrations in $\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{ }\mathit{s}$. Find the angular frequency and the period of motion.

Question: A Gasoline Engine. A gasoline engine takes in \(1.61 \times 1{0^4}\;J\) of heat and delivers \(3700\;J\) of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of  (a) What is the thermal efficiency? (b) How much heat is discarded in each cycle? (c) What mass of fuel is burned in each cycle? (d) If the engine goes through 60.0 cycles per second, what is its power output in kilowatts? In horsepower? 

Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for \(60\) minutes, reaching a point \(2.0{\rm{ km}}\) farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) \(5.0{\rm{ km}}\) downstream from the turnaround point. (a) Assuming a constant paddling effort throughout, how fast is the river flowing?

You drive a car up a steep hill at constant speed. Discuss all of the forces that act on the car. What pushes it up the hill?

A uniform $\mathbf{300}\mathbf{ }\mathit{N}$ trapdoor in a floor is hinged at one side. Find the net upward force needed to begin to open it and the total force exerted on the door by the hinges 

(a) if the upward force is applied at the center and 

(b) if the upward force is applied at the center of the edge opposite the hinges.

The density of gold is \(19.3\,{{\rm{g}} \mathord{\left/

 {\vphantom {{\rm{g}} {{\rm{c}}{{\rm{m}}^{\rm{3}}}}}} \right.

 \kern-\nulldelimiterspace} {{\rm{c}}{{\rm{m}}^{\rm{3}}}}}\). What is this value in kilograms per cubic meter?

Question: A gasoline engine has a power output of \(180\;kW\) (about \(241\;hp\)). Its thermal efficiency is \(28.0\% \). (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

For medical reasons, astronauts in outer space must determine their body mass at regular intervals. Devise a scheme for measuring body mass in an apparently weightless environment.

Raising a Ladder. A ladder carried by a fire truck is$\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$ long. The ladder weighs $\mathbf{3400}\mathbf{ }\mathit{N}$and its center of gravity is at its center. The ladder is pivoted at one end (A) about a pin (Fig. E11.5); ignore the friction torque at the pin. The ladder is raised into position by a force applied by a hydraulic piston at C. Point C is $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$from A, and the force $\stackrel{\mathbf{u}}{\mathbf{F}}$exerted by the piston makes an angle of $40°$with the ladder. What magnitude must $\stackrel{\mathbf{u}}{\mathbf{F}}$have to just lift the ladder off the support bracket at B? Start with a free-body diagram of the ladder.

A machine part is undergoing SHM with a frequency of $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{H}\mathit{z}$ and amplitude $\mathbf{1}\mathbf{.}\mathbf{80}\mathbf{ }\mathit{c}\mathit{m}$. How long does it take the part to go from $\mathit{x}\mathbf{=}\mathbf{0}$ to $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{80}\mathbf{ }\mathit{c}\mathit{m}$?

ln hot-air ballooning, a large balloon is filled with air heated by a gas burner at the bottom. Why must the air be heated? How does the balloonist control ascent and descent?

Two people are carrying a uniform wooden board that is $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{m}$ long and weighs $\mathbf{160}\mathbf{ }\mathit{N}$. If one person applies an upward force equal to $\mathbf{60}\mathbf{ }\mathit{N}$ at one end, at what point does the other person lift? Begin with a free-body diagram of the board. 

The wings of the blue-throated hummingbird (Lampornis clemenciae), which inhabits Mexico and the south-western United States, beat at a rate of up to $\mathbf{900}$ times per minute. Calculate (a) the period of vibration of this bird’s wings, (b) the frequency of the wing’s vibration, and (c) the angular frequency of the bird’s wing beats.

A square field measuring 100.0 m by 100.0 m has an area of 1.00 hectare. An acre has an area of 43,600 ft2 . If a lot has an area of 12.0 acres, what is its area in hectares?

(I) How much work is required to stop an electron \(\left( {m = 9.11 \times {{10}^{ - 31}}\;{\rm{kg}}} \right)\)  which is moving with a speed of \(1.10 \times {10^6}\;{\rm{m/s}}\)?

shows some of the electric field lines due to three-point charges arranged along the vertical axis. All three charges have the same magnitude. (a) What are the signs of the three charges? Explain your reasoning. (b) At what point(s) is the magnitude of the electric field the smallest? Explain your reasoning. Explain how the fields produced by each individual point charge combine to give a small net field at this point or points. 

According to the standard model of the fundamental particles, what are the similarities between baryons and leptons? What are the most important differences?   

When a Dodge Viper is at Elwood’s Car Wash, a BMW Z3 is at Elm and Main. Later, when the Dodge reaches Elm and Main, the BMW reaches Elwood’s Car Wash. How are the cars’ average velocities between these two times related?

The maximum energy that a bone can absorb without breaking depends on characteristics such as its cross-sectional area and elasticity. For healthy human leg bones of approximately $\mathbf{6}\mathbf{.}\mathbf{0}{\mathbf{\text{ cm}}}^{\mathbf{\text{2}}}$ cross-sectional area, this energy has been experimentally measured to be about $\mathbf{200}\mathbf{\text{ J}}$. (a) From approximately what maximum height could a $\mathbf{\text{60 kg}}$ person jump and land rigidly upright on both feet without breaking his legs? (b) You are probably surprised at how small the answer to part (a) is. People obviously jump from much greater heights without breaking their legs. How can that be? What else absorbs the energy when they jump from greater heights? (Hint: How did the person in part (a) land? How do people normally land when they jump from greater heights?) (c) Why might older people be much more prone than younger ones to bone fractures from simple falls (such as a fall in the shower)?

Question: Calculate the energy change required for an electron to move between states: a quantum jump up or down an energy-level diagram.

In Example 8.7 (Section 8.3), where the two gliders ofFig. 8.18 stick together after the collision, the collision is inelastic because ${\mathit{K}}_{\mathbf{1}}\mathbf{<}{\mathit{K}}_{\mathbf{2}}$ . In Example 8.5 (Section 8.2), is the collisioninelastic? Explain

Question: The coefficient of performance K = H/P is a dimensionless quantity. Its value is independent of the units used for H and P, as long as the same units, such as watts, are used for both quantities. However, it is common practice to express H in Btu/h and P in watts. When these mixed units are used, the ratio H/P  is called the energy efficiency ratio (EER). If a room air conditioner has K = 3.0, what is its EER?

A $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{-}\mathit{N}$350-N, uniform, $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{-}\mathit{m}$ 1.50-m bar is suspended horizontally by two vertical cables at each end. Cable A can support a maximum tension of 500.0 N without breaking, and cable B can support up to$\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{-}\mathit{N}$ 400.0 N. You want to place a small weight on this bar. 

(a) What is the heaviest weight you can put on without breaking either cable, and 

(b) where should you put this weight?

A certain fuel-efficient hybrid car gets gasoline mileage of 55.0 mpg (miles per gallon). (a) If you are driving this car in Europe and want to compare its mileage with that of other European cars, express this mileage in km/L 1L = liter2. Use the conversion factors in Appendix E. (b) If this car’s gas tank holds 45 L, how many tanks of gas will you use to drive 1500 km?

56. A uniform disk with a radius \(R = 0.400{\rm{ m}}\) and mass \({\rm{30}}{\rm{.0 kg}}\) rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to \(\theta \left( t \right) = \left( {1.10{{{\rm{ rad}}} \mathord{\left/ {\vphantom {{{\rm{ rad}}} {\rm{s}}}} \right.} {\rm{s}}}} \right)t + \left( {6.30{{{\rm{ rad}}} \mathord{\left/ {\vphantom {{{\rm{ rad}}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right){t^2}\). What is the resultant linear acceleration of a point on the rim of the disk at the instant when the disk has turned through \({\rm{0}}{\rm{.100 rev}}\)?

Engineers are designing a system by which a falling mass \(m\) 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 \(3.71{\rm{ }}{\raise0.7ex\hbox{\({\rm{m}}\)} \!\mathord{\left/

 {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}}\right.\}

\!\lower0.7ex\hbox{\({{{\rm{s}}^{\rm{2}}}}\)}}\) . In the earth tests, when m is set to \(15.0{\rm{ kg}}\)  and allowed to fall through \({\rm{5}}{\rm{.00 m}}\), it gives \(250.0{\rm{ J}}\) of kinetic energy to the drum. (a) If the system is operated on Mars, through what distance would the \(15{\rm{ kg}}\) mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the \(15{\rm{ kg}}\) mass be moving on Mars just as the drum gained \(250.0{\rm{ J}}\) of kinetic energy?

A vacuum cleaner belt is looped over a shaft of radius 0.45 cm and a wheel of radius 1.80 cm. The arrangement of the belt, shaft, and wheel is similar to that of the chain and sprockets in Fig. Q9.4. The motor turns the shaft at 60.0 rev\s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn’t slip on either the shaft or the wheel. 

(a) What is the speed of a point on the belt? 

(b) What is the angular velocity of the wheel, in rad\s?

It has been argued that power plants should make use of off-peak hours (such as late at night) to generate mechanical energy and store it until it is needed during peak load times, such as the middle of the day. One suggestion has been to store the energy in large flywheels spinning on nearly frictionless ball bearings. Consider a flywheel made of iron (density \(7800\,\,{{kg} \mathord{\left/ {\vphantom {{kg} {{m^3}}}} \right.} {{m^3}}}\)) in the shape of a 10.0-cm-thick uniform disk. 

(a) What would the diameter of such a disk need to be if it is to store 10.0 megajoules of kinetic energy when spinning at 90.0 rpm about an axis perpendicular to the disk at its center? 

(b) What would be the centripetal acceleration of a point on its rim when spinning at this rate?

A cylindrical capacitor consists of a solid inner conducting core with radius of 0.250 cm, surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is 12.0 cm. The capacitance is 36.7 pF. 

(a) Calculate the inner radius of the hollow tube. 

(b) When the capacitor is charged to 125 V, what is the charge per unit length l on the capacitor?

A uniform ladder $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{\text{ m}}$ long rests against a frictionless, vertical wall with its lower end $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\text{ m}}$ from the wall. The ladder weighs $\mathbf{160}\mathbf{\text{ N}}$ . The coefficient of static friction between the foot of the ladder and the ground is $\mathbf{040}$. A man weighing $\mathbf{74}\mathit{R}$climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder.

(a) What is the maximum friction force that the ground can exert on the ladder at its lower end? 

(b) What is the actual friction force when the man has climbed $\mathbf{1}\mathbf{.}\mathbf{0}\mathit{m}$along the ladder? 

(c) How far along the ladder can the man climb before the ladder starts to slip? 

Question: A freezer has a coefficient of performance of \(2.40\). The freezer is to convert \(1.80\;kg\) of water at \(25.0^\circ C\) to \(1.80\;kg\) of ice at \( - 5.0^\circ C\) in one hour. (a) What amount of heat must be removed from the water at \(25.0^\circ C\) to convert it to ice at \( - 5.0^\circ C\)? (b) How much electrical energy is consumed by the freezer during this hour? (c) How much wasted heat is delivered to the room in which the freezer sits?

A water wave traveling in a straight line on a lake is described by the equation

               \(y\left( {x,t} \right) = \left( {2.75\,cm} \right)cos\left( {0.410\,{{rad} \mathord{\left/

 {\vphantom {{rad} {cm}}} \right.

 \kern-\nulldelimiterspace} {cm}}\,x + 6.20{{rad} \mathord{\left/

 {\vphantom {{rad} s}} \right.

 \\,t} \right)\) 

Where \(y\) is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

In uniform circular motion, how does the acceleration change when the speed is increased by a factor of 3? When the radius is decreased by a factor of 2?

A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of 0.0200 N at that point. (a) Find its angular acceleration. (b) How long will it take to decrease its rotational speed by 22.5 rad/s?

A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end (Fig. E 11.11). The diving board is of uniform cross section and weighs 280 N. Find :

(a) The force at the support point and 

(b) The force at the left-hand end.

A relaxed biceps muscle requires a force of 25.0 N for an elongation of ; 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 50.0 cm².

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