Mechanics

A metal rod that is 4.00 m long and $\mathbf{0}\mathbf{.}\mathbf{50}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{2}}$ 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?

Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1 degrees from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniel’s velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

Two very long insulated wires perpendicular to each other in the same plane carry currents as shown in Fig. Find the magnitude of the net magnetic field these wires produce at points P and Q if the 10.0A current  to the left.

According to the shop manual, when drilling should have a 12.7-mm-diameter in wood, plastic, or aluminum a drill should have a speed of 1250 rev/min. For a 12.7-mm-diameter  drill bit turning at a constant 1250 rev/min .find 

a) the maximum linear speed of any part of the bit and

Question: A stockroom worker pushes a box with mass 16.8 kg on a horizontal surface with a constant speed of 3.50m/s. The coefficient of kinetic friction between the box and the surface is 0.20. (a) What horizontal force must the worker apply to maintain the motion? (b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest?

A 22-caliber riffle bullet traveling at $\mathbf{350}\mathbf{m}\mathbf{/}\mathbf{s}$ strikes a large tree and penetrates it to a depth of $\mathbf{0}\mathbf{.}\mathbf{130}\mathbf{m}$. The mass of the bullet is $\mathbf{1}\mathbf{.}\mathbf{80}\mathbf{g}$. Assume a constant retarding force. (a) How much time is required for the bullet to stop? (b) What force, in Newton, does the tree exert on the bullet?

A gyroscope takes 3.8 s to precess 1.0 revolution about a vertical axis. Two minutes later, it takes only 1.9 s to precess 1.0 revolution. No one has touched the gyroscope. Explain.

A nylon rope used by mountaineers elongates 1.10 m 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?

Four small spheres, each of which you can regard as a point of mass of 0.200 kg, are arranged in a square of 0.400 m on a side and connected by extremely light rods (Fig. E9.28). Find the moment of inertia of the system about an axis

 through the center of the square, perpendicular to its plane (an axis through point O in the figure);

 bisecting two opposite sides of the square (an axis along the line AB in the figure);

 that passes through the centers of the upper left and lower right spheres and point O.

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is\(y\left( {x,t} \right) = 2.30\,mm\,cos\left[ {\left( {6.98\,{{rad} \mathord{\left/{\vphantom {{rad} m}} \right.

 \nulldelimiterspace} m}} \right)x + \left( {742\,{{rad} \mathord{\left/{\vphantom {{rad} s}} \right.

\nulldelimiterspace} s}} \right)t} \right]\). Being more practical, you measure the rope to have a length of \(1.35 m\) and a mass of\(0.00338 kg\). You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Let  $\mathbf{\theta }$ be the angle that the vector  $\overrightarrow{\mathbf{A}}$ makes with the +x-axis, measured counterclockwise from that axis. Find angle $\mathbf{\theta }$  for a vector that has these components: (a) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}\mathbf{ }{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$ ; (b) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$ ; (c) ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$ ; (d)  ${\mathbf{A}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{,}{\mathbf{A}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$. 

You (mass $\mathbf{55}\mathbf{kg}$) are riding a frictionless skateboard (mass $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{kg}$) in a straight line at a speed of $\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$. A friend standing on a balcony above you drops a $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\text{ kg}}$ sack of flour straight down into your arms. (a) What is your new speed while you hold the sack? (b) Since the sack was dropped vertically, how can it affect your horizontal motion? Explain. (c) Now you try to rid yourself of the extra weight by throwing the sack straight up. What will be your speed while the sack is in the air? Explain.

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 $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{2}}$ 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 10.0 kg. For each wire, compute (a) the tensile strain and (b) the elongation.

An astronaut in space cannot use conventional means, such as a scale or balance, to determine the mass of an object. But she does have devices to measure distance and time accurately. She knows her own mass is $\mathbf{78}\mathbf{.}\mathbf{4}\mathbf{\text{ kg}}$, but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at $\mathbf{3}\mathbf{.}\mathbf{50}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$, she pushes against it, which slows it down to $\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$ (but does not reverse it) and gives her a speed of $\mathbf{2}\mathbf{.}\mathbf{40}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$. What is the mass of this canister?

Vector A is in the direction 34.0 clockwise from the -yaxis. The x-component of A is Ax = -16.0 m. (a) What is the y-component of A ? (b) What is the magnitude of A ?

Reflection. A wave pulse on a string has the dimensions shown in Fig. E15.31 at \(t = 0\). The wave speed is \(5\;{{\rm{m}} \mathord{\left/

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

 \nulldelimiterspace} {\rm{s}}}\). (a) If point \(O\) is a fixed end, draw the total wave on the string at \(t = 1.0\;{\rm{ms}}\), \(2.0\;{\rm{ms}}\), \(3.0\;{\rm{ms}}\), \(4.0\;{\rm{ms}}\), \(5.0\;{\rm{ms}}\), \(6.0\;{\rm{ms}}\), and \(7.0\;{\rm{ms}}\). (b) Repeat part (a) for the case in which point \(O\) is a free end

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass $7.50\text{ kg}$ is sliding to the left at $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$, while the other, of mass , is slipping to the right at $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \text{s}}$}\right.$. They hold fast to each other after they collide. (a) Find the magnitude and direction of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?

(a) How much work must be done on a particle with mass m to accelerate it (a) from rest to a speed of 0.090c and (b) from a speed of 0.900c to a speed of 0.990c? (Express the answers in terms of mc^2 .) (c) How do your answers in parts (a) and (b) compare?

An average sleeping person metabolizes at a rate of about $80 \mathrm{W}$ by digesting food or burning fat. Typically, $20\%$ of this energy goes into bodily functions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all this excess heat by transferring it (by conduction and the flow of blood) to the surface of the body, where it is radiated away. The normal

internal temperature of the body (where the metabolism takes place) is $37°\mathrm{C}$, and the skin is typically  $7°\mathrm{C}$ cooler. By how much does the person’s entropy change per second due to this heat transfer?

Consider the ring-shaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use  to find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and  when ? Explain why these results make sense. 

Write each vector in Fig. E1.24 in terms of the unit vectors $\widehat{\mathbf{i}}$  and  $\widehat{j}$

A proud deep-sea fisherman hangs a 65.0-kg fish from an ideal spring having negligible mass. The fish stretches the spring 0.180 m. (a) Find the force constant of the spring. The fish is now pulled down 5.00 cm and released. (b) What is the period of oscillation of the fish? (c) What is the maximum speed it will reach?

Exercises 10.39: A hollow, thin-walled sphere of mass $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{\text{ kg}}$ and diameter $\mathbf{48}\mathbf{.}\mathbf{0}\mathbf{\text{ cm}}$ is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by $\mathit{\theta }\left(t\right)\mathbf{=}\mathit{A}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathit{B}{\mathit{t}}^{\mathbf{4}}$, where $\mathit{A}$ has numerical value $\mathbf{1}\mathbf{.}\mathbf{50}$ and $\mathit{B}$  has numerical value $\mathbf{1}\mathbf{.}\mathbf{10}$. (a) What are the units of the constants  $\mathit{A}$ and $\mathit{B}$? (b) At the time $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\text{ s}}$, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

A hollow, thin-walled sphere of mass $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{\text{ kg}}$ and diameter  $\mathbf{48}\mathbf{.}\mathbf{0}\mathbf{\text{ cm}}$ is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by $\mathit{\theta }\left(t\right)\mathbf{=}\mathit{A}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathit{B}{\mathit{t}}^{\mathbf{4}}$, where $\mathit{A}$ has numerical value $\mathbf{1}\mathbf{.}\mathbf{50}$  and   has numerical value $\mathbf{1}\mathbf{.}\mathbf{10}$. (a) What are the units of the constants $\mathit{A}$ and $\mathit{B}$? (b) At the time $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\text{ s}}$, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

To study damage to aircraft that collide with large birds, you design a test gun that will accelerate chicken-sized objects so that their displacement along the gun barrel is given by $\mathbf{x}\mathbf{=}\left(9.0\times {10}^3 m/s^2\right){\mathbf{t}}^{\mathbf{2}}\mathbf{-}\left(8.0\times {10}^{4}m/s^2\right){\mathbf{t}}^{\mathbf{3}}$. The object leaves the end of the barrel at t=0.025 s. (a) How long must the gun barrel be? (b) What will be the speed of the objects as they leave the end of the barrel? (c) What net force must be exerted on a 1.50-kg  object at (i) t=0  and (ii) t=0.025 s ?

In Fig. 29.23 the capacitor plates have area \({\bf{5}}.{\bf{00}}\,{\bf{c}}{{\bf{m}}^2}\)and separation\({\bf{2}}.{\bf{00}}\,{\bf{mm}}\). The plates are in vacuum. The charging current \({i_{\rm{C}}}\) has a constant value of \({\bf{1}}.{\bf{80}}\,{\bf{mA}}\). At \(t = {\bf{0}}\) the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when \(t = {\bf{0}}.{\bf{500}}\,{\rm{\mu }}{\bf{s}}\) . (b) Calculate \({\bf{dE}}/{\bf{dt}}\), the time rate of change of the electric field between the plates. Does \({\bf{dE}}/{\bf{dt}}\) vary in time? (c) Calculate the displacement current density \({j_{\rm{D}}}\) between the plates, and from this the total displacement current \({i_{\rm{D}}}\). How do \({i_{\rm{C}}}\) and \({i_{\rm{D}}}\) compare?

Find the vector product $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}\mathbf{\times }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}$ (expressed in unit vectors) of the two vectors given in Exercise 1.38. What is the magnitude of the vector product?

Question: Find the vector product $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{B}}$ (expressed in unit vectors)

of the two vectors given in Exercise 1.38. What is the magnitude

of the vector product?

A circular coil with area \(A\) and \(N\) turns is free to rotate about a diameter that coincides with the \(x\)axis. Current \(I\)  is circulating in the coil. There is a uniform magnetic field \(\vec B\)in the positive \(y\) direction. Calculate the magnitude and direction of the torque \(\vec \tau \)  and the value of the potential energy \(U\) as given in Eq. (27.27), when the coil is oriented as shown in parts (a) through (d) of Fig. E27.45.

 Find the angle between each of these pairs of vectors:

(a) $\overrightarrow{A }=-2.00i \mathit{+}\mathit{ }\mathit{6}\mathit{.}\mathit{00}j$ and $\overrightarrow{B}=2.00\hat{i}$$-3.00\hat{j}$

(b) $\overrightarrow{A}= 3.00\hat{i}+5.00\hat{j}$ and $\overrightarrow{B}=10.00\hat{i}$$+6.00\hat{j}$

(c) $\overrightarrow{A}=-4.00\hat{i}$$+2.00\hat{j}$and $\overrightarrow{B}= 7.00\hat{i}$$+14.00\hat{j}$

Find the angle between each of these pairs of vectors:

(a) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{6}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}\mathbf{ }\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}$ 

(b)$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{5}\mathbf{.}\mathbf{00}\widehat{\mathbf{j}\mathbf{ }}\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{6}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}$

(c) $\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{00}\widehat{\mathbf{j}\mathbf{ }}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{ }\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{14}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}$ 

  A metal sphere with radius $r_a=1.20 cm$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b=9.60 cm$. Charge $\mathbf{+}\mathbf{q}$ is put on the inner sphere and charge -q on the outer spherical shell. The magnitude of $\mathbf{q}$ is chosen to make the potential difference between the spheres $\mathbf{500}\mathbf{V}$ with the inner sphere at higher potential. (a) Use the result of Exercise $\mathbf{23}\mathbf{.}\mathbf{41}\left(b\right)$to calculate $\mathbf{q}$. $\left(b\right)$ With the help of the result of Exercise $\mathbf{23}\mathbf{.}\mathbf{41}\left(a\right)$,sketch the equipotential surfaces that correspond to $\mathbf{500}\mathbf{,}\mathbf{ }\mathbf{400}\mathbf{,}\mathbf{ }\mathbf{300}\mathbf{,}\mathbf{ }\mathbf{200}\mathbf{,}\mathbf{ }\mathbf{100}\mathbf{,}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{0}\mathbf{ }\mathbf{V}\mathbf{.}$$\left(c\right)$ In your sketch, show the electric field lines. Are the electric field lines and equipotential surfaces mutually perpendicular? Are the equipotential surfaces closer together when the magnitude of $\overrightarrow{E}$ is largest?

A coil with magnetic moment \(1.45\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}}\)  is oriented initially with its magnetic moment anti-parallel to a uniform \(0.835\;{\rm{T}}\) magnetic field. What is the change in potential energy of the coil when it is rotated \(180^\circ \) so that its magnetic moment is parallel to the field?

Question: In a shunt-wound dc motor with the field coils and rotor connected in parallel (Fig. E27.47), the resistance \({R_f}\) of the field coils is \(106\;\Omega \), and the resistance \({R_r}\) of the rotor is \(5.9\;\Omega \). When a potential difference of \(120\;V\) is applied to the brushes and the motor is running at full speed delivering mechanical power, the current supplied to it is \(4.82\;A\). 

(a) What is the current in the field coils? 

(b) What is the current in the rotor?

(c) What is the induced emf developed by the motor? 

(d) How much mechanical power is developed by this motor?

A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of $\mathbf{40}.\mathbf{0}°$above the horizontal. The glider has mass $\mathbf{0}\mathbf{.}\mathbf{0900}\mathbf{ }\mathbf{kg}$ The spring has $\mathit{k}\mathbf{=}\mathbf{640}\mathbf{ }\mathbf{\text{N/m}}$ negligible mass. When the spring is released, the glider travels a maximum distance of $\mathbf{1}\mathbf{.}\mathbf{80}\mathbf{ }\mathbf{m}$ along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. (a) What distance was the spring originally compressed? (b) When the glider has traveled along the air track $\mathbf{0}.\mathbf{80} \mathbf{m}$ from its initial position against the compressed spring, is it still in contact with the spring? What is the kinetic energy of the glider at this point?

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant $\mathit{k}\mathbf{=}\mathbf{450}\mathbf{ }\mathbf{\text{N/m}}$and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass $\mathbf{1}.\mathbf{80} \mathbf{kg}$ and is to reach a maximum height of $\mathbf{3}.\mathbf{6} \mathbf{m}$ above its initial position on the compressed spring, what distance must the bricklayer compress the spring initially? (The brick loses contact with the spring when the spring returns to its uncompressed length. Why?)

A force in the $\mathbf{+}\mathit{x}$  direction with magnitude $\mathit{F}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{N}}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{530}\mathbf{ }\mathbf{\text{N/m)}}\mathit{x}$ is applied to a $\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{kg}}$ box that is sitting on the horizontal, frictionless surface of a frozen lake. $\mathit{F}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is the only horizontal force on the box. If the box is initially at rest at $\mathit{x}\mathbf{=}\mathbf{0}$ , what is its speed after it has traveled $\mathbf{14}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{m}}$ ?

A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of $\mathit{a}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{80}\mathbf{ }{\mathbf{\text{m/s}}}^{\mathbf{\text{2}}}$ . A worker assists the cart by pushing on the crate with a force that is eastward and has magnitude that depends on time according to $\mathit{F}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\left(5.40 \text{N/s}\right)\mathit{t}$ . What is the instantaneous power supplied by this force at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{s}}$?

A 1500-kg rocket is to be launched with an initial upward speed of 50.0 m/s. In order to assist its engines, the engineers will start it from rest on a ramp that rises \(53^\circ \) above the horizontal (Fig. P7.50). At the bottom, the ramp turns upward and launches the rocket vertically. The engines provide the constant forward thrust of 2000 N, and friction with the ramp surface is a constant 500 N. How far from the base of the ramp should the rocket start, as measured along the surface of the ramp?

Question: In the laboratory, a student studies a pendulum by graphing the angle \(\theta \) that the string makes with the vertical as a function of time \(t\), obtaining the graph shown in Fig. E14.50. (a) What are the period, frequency, angular frequency, and amplitude of the pendulum’s motion? (b) How long is the pendulum? (c) Is it possible to determine the mass of the bob?

Question: The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00m long, and the upper end of the cable is fastened to the arm at a point 3.00m from the central shaft (Fig. E5.50). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of 30.0^o with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

How many joules of energy does a $\mathbf{100}\mathbf{\text{ watt}}$ light bulb use per hour? How fast would a $\mathbf{70}\mathbf{\text{ kg}}$  person have to run to have that amount of kinetic energy?

Electromagnetic radiation is emitted by accelerating charges. The rate at which energy is emitted from an accelerating charge that has charge q and acceleration a is given by $\frac{\mathbf{dE}}{\mathbf{dt}}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}{\mathbf{a}}^{\mathbf{2}}}{\mathbf{6}{\mathbf{\tau \tau \epsilon }}_{\mathbf{0}}{\mathbf{c}}^{\mathbf{3}}}$ where c is the speed of light. (a) Verify that this equation is dimensionally correct. (b) If a proton with a kinetic energy of 6.0 MeV is traveling in a particle accelerator in a circular orbit of radius 0.750 m, what fraction of its energy does it radiate per second? (c) Consider an electron orbiting with the same speed and radius. What fraction of its energy does it radiate per second?

In another version of the “Giant Swing” (see Exercise 5.50), the seat is connected to two cables, one of which is horizontal (Fig. E5.51). The seat swings in a horizontal circle at a rate of . If the seat weighs and an person is sitting in it, find the tension in each cable.

A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

BIO Should You Walk or Run? It is $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{\text{ km}}$ from your home to the physics lab. As part of your physical fitness program, you could run that distance at $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{$\mathbf{k}\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{h}$}\right.$  (which uses up energy at the rate of $\mathbf{700}\mathbf{\text{ W}}$), or you could walk it leisurely at $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{$\mathbf{k}\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{h}$}\right.$ (which uses energy at $\mathbf{290}\mathbf{\text{ W}}$). Which choice would burn up more energy, and how much energy (in joules) would it burn? Why does the more intense exercise burn up less energy than the less intense exercise?

Magnetar. On December 27, 2004, astronomers observed the greatest flash of light ever recorded from outside the solar system. It came from the highly magnetic neutron star SGR 1806-20 (a magnetar). During $\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{\text{ s}}$, this star released as much energy as our sun does in  $\mathbf{250}\mathbf{,}\mathbf{000}\mathbf{\text{ years}}$. If $\mathit{P}$ is the average power output of our sun, what was the average power output (in terms of $\mathit{P}$) of this magnetar?

The acceleration of a motorcycle is given by ${\mathbf{a}}_{\mathbf{x}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{At}\mathbf{-}{\mathbf{Bt}}^{\mathbf{2}}$, where A = $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}$  and B = $\mathbf{0}\mathbf{.}\mathbf{120}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{4}}$. The motorcycle is at rest at the origin at time . (a) Find its position and velocity as functions of time. (b) Calculate the maximum velocity it attains.

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section 39.3), in the lowest energy state the electron orbits the proton at a speed of 6 2.2 10 m/s in a circular orbit of radius 5.3 10 m. (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current I? (c) What is the magnetic moment of the atom due to the motion of the electron?