Measurement

A single force acts on a 3.0 kg particle-like object whose position is given by $\mathbf{x}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{t}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}{\mathbf{t}}^{\mathbf{3}}$, with x in meters and t in seconds. Find the work done by the force from $\mathbf{t}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{t}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$.

As two trains move along a track, their conductors suddenly notice that they are headed towards each other. Figure 2-31 gives their velocities v  as functions of time t as the conductors slow the trains. The figure’s vertical scaling is set by ${\mathit{v}}_{\mathbf{s}}\mathbf{=}\mathbf{40}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$. The slowing processes begin when the trains are 200 m apart. What is their separation when both trains have stopped?

Figure 7-41 shows a cord attached to a cart that can slide along a frictionless horizontal rail aligned along an x axis. The left end of the cord is pulled over a pulley, of negligible mass and friction and at cord height $\mathbf{h}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{m}$, so the cart slides from ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{ }\mathbf{to}\mathbf{ }{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$. During the move, the tension in the cord is a constant $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{N}$. What is the change in the kinetic energy of the cart during the move?

In Figure, a battery of potential difference V=12 V is connected to a resistive strip of resistance $\mathit{R}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\Omega }$ .When an electron moves through the strip from one end to the other, (a) In which direction in the figure does the electron move,(b) How much work is done on the electron by the electric field in the strip, and (c) How much energy is transferred to the thermal energy of the strip by the electron?

Question: Suppose that a simple pendulum consists of a small 60.0 g  bob at the end of a cord of negligible mass. If the angle between the cord and the vertical is given by, 

$\mathit{\theta }\mathbf{=}\left(0.0800\text{rad}\right)\mathbf{\text{cos}}\left[\left(4.43\text{rad}/\text{s}\right)t+\right]$ ,

1. What is the pendulum’s length?
2. What is its maximum kinetic energy?

Humid air brakes down (its molecules become ionized) in an electric field of 3.0 X ${10}^{6 }N\backslash C$. In that field, what is the magnitude of the electrostatic force on (a) an electron and (b) an ion with a single electron missing?

Question: Suppose that the radius of the Sun was increased to 5.9010 ¹² m (the average radius of the orbit of Pluto), that the density of this expanded Sun were uniform, and that the planets revolved within this tenuous object. (a) Calculate Earth’s orbital speed in this new configuration. (b) What is the ratio of the orbital speed calculated in (a) to Earth’s present orbital speed of 29.8 km/s? Assume that the radius of Earth’s orbit remains unchanged. (c) What would be Earth’s new period of revolution? (The Sun’s mass remains unchanged).

A force of 5.0 N acts on a 15.0 kg body initially at rest. Compute the work done by the force in (a) the first, (b) the second, and (c) the third seconds and (d) the instantaneous power due to the force at the end of the third second.

Question: In the Figure below, a meter stick initially swings about a pivot point at one end, at distance h  from the stick’s center of mass.

1. If this physical pendulum is inverted and suspended at point P , what is its new period of oscillation?
2. Is the period now greater than, less than, or equal to its previous value?

A skier is pulled by a towrope up a frictionless ski slope that makes an angle of${\mathbf{12}}^{\mathbf{°}}$with the horizontal. The rope moves parallel to the slope with a constant speed of $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$. The force of the rope does $\mathbf{900}\mathbf{ }\mathbf{J}$ of work on the skier as the skier moves a distance of $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$ up the incline. (a) If the rope moved with a constant speed of $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$, how much work would the force of the rope do on the skier as the skier moved a distance of $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$up the incline? At what rate is the force of the rope doing work on the skier when the rope moves with a speed of (b) $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and (c) $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$?

Four identical particles of mass 0.50kg each are placed at the vertices of a $\mathbf{2}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{\times }\mathbf{2}\mathbf{.}\mathbf{0}\mathit{m}$ square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

A 100 kg block is pulled at a constant speed of 5.0 m/s across a horizontal floor by an applied force of 122 N directed ${\mathbf{37}}^{\mathbf{°}}$ above the horizontal. What is the rate at which the force does work on the block?

The loaded cab of an elevator has a mass of $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }\mathbf{kg}$ and moves 210 m up the shaft in 23 s at constant speed. At what average rate does the force from the cable do work on the cab?

Question: Figure 44-12 is a hypothetical plot of the recessional speeds v of galaxies against their distance r from us; the best-fit straight line through the data points is shown. From this plot determine the age of the universe, assuming that Hubble’s law holds, and that Hubble’s constant has always had the same value.

Visible light is incident perpendicularly on a grating with 315 rulings/mm. What is the longest wavelength that can be seen in the fifth-order diffraction?

A machine carries a 4.0 kg package from an initial position of $\overrightarrow{{\mathbf{d}}_{\mathbf{i}}}\mathbf{=}\left(0.50 m\right)\hat{\mathbf{i}}\mathbf{+}\left(0.75 m\right)\hat{\mathbf{j}}\mathbf{+}\left(0.20 m\right)\hat{\mathbf{k}}$ at t=0 to a final position of $\overrightarrow{{\mathbf{d}}_{\mathbf{f}}}\mathbf{=}\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{50}\mathbf{ }\mathbf{m}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{m}\mathbf{)}\hat{\mathbf{k}}$ at t=12 s. The constant force applied by the machine on the package is $\overrightarrow{\mathbf{F}}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{N}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{N}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{N}\mathbf{)}\hat{\mathbf{k}}$. For that displacement, find (a) the work done on the package by the machine’s force and (b) the average power of the machine’s force on the package.

A heating element is made by maintaining a potential difference of 75.0V across the length of a Nichrome wire that has a $\mathbf{2}\mathbf{.}\mathbf{60}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}{\mathbf{m}}^{\mathbf{2}}$ cross section. Nichrome has a resistivity of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{\Omega m}$.(a) If the element dissipates, what is its length?(b) If 100 V is used to obtain the same dissipation rate, what should the length be?

The current density $\overrightarrow{\mathbf{J}}\mathbf{ }$  inside a long, solid, cylindrical wire of radius a= 3.1mm is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to $\mathit{J}\mathbf{=}\mathbf{ }{\mathit{J}}_{\mathbf{0}}\mathit{r}\mathbf{/}\mathit{a}\mathbf{,}$  where ${\mathit{J}}_{\mathbf{0}}\mathbf{=}\mathbf{310}\mathbf{ }\mathit{A}\mathbf{/}{\mathit{m}}^{\mathbf{2}}$.  (a) Find the magnitude of the magnetic field at $\mathit{r}\mathbf{=}\mathbf{ }\mathbf{0}$,  (b) Find the magnitude of the magnetic field  $\mathit{r}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{a}\mathbf{/}\mathbf{2}$, and(c) Find the magnitude of the magnetic field $\mathit{r}\mathbf{=}\mathbf{ }\mathit{a}$ .

Light of wavelength 121.6 nm is emitted by a hydrogen atom. What are the (a) higher quantum number and (b) lower quantum number of the transition producing this emission? (c) What is the series that includes the transition? 

A hoodlum throws a stone vertically downward with an initial speed of $\mathbf{12}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ from roof of a building, 30 m above the ground. (a) How long does it take the stone to reach the ground? (b) What is the speed of stone at impact?

During the launch from a board, a diver’s angular speed about her center of mass changes from zero to 6.20rad/s in 220ms. Her rotational inertia about her center of mass is 12.0kg.m² . During the launch, what are the magnitudes of (a) her average angular acceleration and (b) the average external torque on her from the board?

Question: At time t=0, apple 1 is dropped from a bridge onto a roadway beneath the bridge; somewhat later, apple 2 is thrown down from the same height. Figure 2-33 gives the vertical positions y of the apples versus t during the falling, until both apples have hit the roadway. The scaling is set by t_s=2s. With approximately what speed is apple 2 thrown down?

Wire C and wire D are made from different materials and have length ${\mathit{L}}_{\mathbf{c}}\mathbf{=}{\mathit{L}}_{\mathbf{D}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$.  The resistivity and diameter of wire C are $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{\Omega }\mathbf{.}\mathbf{m}$ and 1.00 mm, and those of wire D are . 0.25 mm and 0.50 mm. The wires are joined as shown in Figure, and a current of 2.0 A is set up in them. What is the electric potential difference between (a) points 1 and 2 and (b) points 2 and 3? What is the rate at which energy is dissipated between (c) points 1 and 2 and (d) points 2 and 3?

A 120 V potential difference is applied to a space heater that dissipates 500 W  during operation. (a) What is its resistance during operation? (b) At what rate do electrons flow through any cross section of the heater element?

55, 57 53 Thin lenses. Object $\mathit{O}$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table 34-6 gives object distance p (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $\mathit{i}$  and (b) the lateral magnification m of the object, including signs. Also, determine whether the image is (c) real  $\mathbf{\left(}\mathit{R}\mathbf{\right)}$ or virtual $\mathbf{\left(}\mathit{V}\mathbf{\right)}$ , (d) inverted $\mathbf{\left(}\mathit{I}\mathbf{\right)}$  from object $\mathit{O}$ or noninverted  $\left(NI\right)$, and (e) on the same side of the lens as object $\mathit{O}$ or on the opposite side.

Question: In a damped oscillator with , m = 250 g , k = N/m and b = 70 g / s, what is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles?

A force ${\overrightarrow{\mathit{F}}}_{\mathit{a}}$ is applied to a bead as the bead is moved along a straight wire through displacement $\mathit{+}\mathit{5}\mathit{.}\mathit{0}\mathit{c}\mathit{m}$. The magnitude of  ${\overrightarrow{\mathit{F}}}_{\mathit{a}}$ is set at a certain value, but the $\mathit{\varphi }$ angle ${\overrightarrow{\mathit{F}}}_{\mathit{a}}$ between and the bead’s displacement can be chosen. Figure $\mathbf{7}\mathbf{-}\mathbf{45}$ gives the work $\mathit{W}$ done by  on the bead for a range of $\mathit{\varphi }$ values;  ${\mathit{W}}_{\mathit{0}}\mathit{=}\mathit{ }\mathit{25}\mathbf{ }\mathbf{J}$. How much work is done by  ${\overrightarrow{\mathbf{F}}}_{\mathbf{a}}\mathbf{ }$if $\mathit{\varphi }$ is (a) $\mathbf{64}\mathbf{°}$and (b) $\mathbf{147}\mathbf{°}$?

What is the magnitude of the acceleration of a sprinter running at  10 m/s when rounding a turn of radius 25 m ?

Use the results displayed in Problem 61 to predict the (a) magnitude and (b) inclination of Earth’s magnetic field at the geomagnetic equator, the (c) magnitude and (d) inclination at geomagnetic latitude $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{°}$, and the (e) magnitude and (f) inclination at the north geomagnetic pole.

Question: A drowsy cat spots a flowerpot that sails first up and then down past an open window. The pot is in view for a total of 0.50 s, and the top to bottom height of the window is 2.00 m. How high above the window top does the flowerpot go?

A 2.0 kW heater element from a dryer has a length of 80 cm . If a 10 cm section is removed, what power is used by the now shortened element at 120 V ?

A proton and an electron form two corners of an equilateral triangle of side length $\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathit{m}$.What is the magnitude of the net electric field these two particles produce at the third corner?

An iceboat is at rest on a frictionless frozen lake when a sudden wind exerts a constant force of 200 N, toward the east, on the boat. Due to the angle of the sail, the wind causes the boat to slide in a straight line for a distance of 8.0 m in a direction 20 north of east. What is the kinetic energy of the iceboat at the end of that 8.0 m?

By measuring the go-and-return time for a laser pulse to travel from an Earth-bound observatory to a reflector on the Moon, it is possible to measure the separation between these bodies. (a) What is the predicted value of this time? (b) The separation can be measured to a precision of about 15 cm. To what uncertainty in travel time does this correspond? (c) If the laser beam forms a spot on the Moon 3 km in diameter, what is the angular divergence of the beam?

A certain capacitor is charged to a potential difference$\mathit{V}$. If you wish to increase its stored energy by 10%, by what percentage should you increase$\mathit{V}$?

A certain gyroscope consists of a uniform disk with a $\mathbf{50}\mathbf{ }\mathbf{cm}\mathbf{ }$ radius mounted at the centre of an axle that is $\mathbf{11}\mathbf{ }\mathbf{cm}\mathbf{ }$  long and of negligible mass. The axle is horizontal and supported at one end. If the disk is spinning around the axle at  $\mathbf{1000}\mathbf{ }\mathit{r}\mathit{e}\mathit{v}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}$, what is the precession rate?

Figure 32-38 shows a loop model (loop $\mathit{L}$) for a paramagnetic material. (a) Sketch the field lines through and about the material due to the magnet. What is the direction of (b) the loop’s net magnetic dipole moment $\overrightarrow{\mathbf{\mu }}$, (c) the conventional current $\mathit{i}$ in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force acting on the loop?

A thin-walled pipe rolls along the floor. What is the ratio of its translational kinetic energy to its rotational kinetic energy about the central axis parallel to its length?

Question: A particle moves along a straight path through displacement $\overrightarrow{d}\mathbf{=}\left(\mathbf{8}\mathbf{.}\mathbf{0} \mathbf{m}\right)\hat{i}\mathbf{+}\mathbf{c}\hat{j}$while force$\overrightarrow{F}\mathbf{=}\left(\mathbf{2}\mathbf{.}\mathbf{0} \mathbf{N}\right)i\mathbf{+}\left(\mathbf{4}\mathbf{.}\mathbf{0} \mathbf{N}\right)j$ acts on it. (Other forces also act on the particle.) What is the value of c if the work done by $\overrightarrow{\mathbf{F}}$on the particle is (a) zero, (b) positive, and (c) negative?

Question: In Fig. 27-72, the ideal batteries have emfs${\epsilon }_1=20.0V, {\epsilon }_2=10.0V, and {\epsilon }_3=5.0V$, and the resistances are each $2.00\Omega$ . What are the (a) size and (b) direction (left or right) of current $i_1$? (c) Does battery 1 supply or absorb energy, and (d) what is its power? (e) Does battery 2 supply or absorb energy, and (f) what is its power? (g) Does battery 3 supply or absorb energy, and (h) what is its power?

In Fig. 30-71, the battery is ideal and ${\mathbf{\epsilon }}_{\mathbf{battery}}\mathbf{=}\mathbf{10}\mathbf{ }\mathbf{V}\mathbf{ }\mathbf{,}\mathbf{ }{\mathbf{R}}_{\mathbf{1}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{\Omega }\mathbf{ }\mathbf{,}{\mathbf{R}}_{\mathbf{2}}\mathbf{=}\mathbf{10}\mathbf{\Omega }\mathbf{,}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{L}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{H}$ . Switch S is closed at time t = 0. Just afterwards, what are (a),${\mathit{I}}_{\mathbf{1}}$(b) ${\mathit{I}}_{\mathbf{2}}$ (c) the current is through the switch, (d) the potential difference ${\mathit{V}}_{\mathbf{2}}$ across resistor 2, (e) the potential difference ${\mathit{V}}_{\mathbf{L}}$ across the inductor, and (f) the rate of change $\mathit{d}{\mathit{I}}_{\mathbf{2}}\mathbf{/}\mathit{d}\mathit{t}$? A long time later, what are (g)${\mathit{I}}_{\mathbf{1}}$, (h)${\mathit{I}}_{\mathbf{2}}$, (i) ${\mathit{I}}_{\mathit{s}}$, (j) ${\mathit{V}}_{\mathbf{2}}$, (k) ${\mathit{V}}_{\mathbf{L}}$, and (l)$\mathit{d}{\mathit{I}}_{\mathbf{2}}\mathbf{/}\mathit{d}\mathit{t}$ ?

An object is tracked by a radar station and determined to have a position vector given by$\overrightarrow{\mathbf{r}}\mathbf{=}\mathbf{(}\mathbf{3500}\mathbf{-}\mathbf{160}\mathbf{t}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{2700}\hat{\mathbf{j}}\mathbf{+}\mathbf{300}\hat{\mathbf{k}}\mathbf{ }\mathbf{,}\mathbf{with}\mathbf{ }\overrightarrow{\mathbf{r}}$in meters and t in seconds. The radar station’s x axis points east, its y axis north, and its z axis vertically up. If the object is a 250 kg meteorological missile, what are (a) its linear momentum,  (b) its direction of motion, and  (c) the net force on it?

A particle of mass $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{g}}$moves at   $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\raisebox{1ex}{$\mathbf{\text{km}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{s}}$}\right.$in an $\mathit{x}\mathit{y}$ plane, in a region with a uniform magnetic field given by $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\widehat{\mathbf{\text{i}}}\mathbf{\text{mT}}$ . At one instant, when the particle’s velocity is directed${\mathbf{37}}^{\mathbf{\text{o}}}$  counterclockwise from the positive direction of the x axis, the magnetic force on the particle is $\mathbf{0}\mathbf{.}\mathbf{48}\hat{\mathbf{k}}\mathbf{\text{N}}$ .What is the particle’s charge?

Figure $\mathbf{10}\mathbf{-}\mathbf{54}$shows a flat construction of two circular rings that have a common center and are held together by three rods of negligible mass. The construction, which is initially at rest, can rotate around the common center (like a merry-go-round), where another rod of negligible mass lies. The mass, inner radius, and outer radius of the rings are given in the following table. A tangential force of magnitude $12.0N$is applied to the outer edge of the outer ring for $0.300s$.What is the change in the angular speed of the construction during the time interval?

In Fig. 30-75a, switch S has been closed on A long enough to establish a steady current in the inductor of inductance ${\mathit{L}}_{\mathbf{1}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{mH}$and the resistor of resistance ${\mathit{R}}_{\mathbf{1}}\mathbf{=}\mathbf{25}\mathbf{.}\mathbf{0}$. Similarly, in Fig. 30-75b, switch S has been closed on A long enough to establish a steady current in the inductor of inductance ${\mathit{L}}_{\mathbf{2}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{mH}$and the resistor of resistance ${\mathit{R}}_{\mathbf{2}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{\Omega }$. The ratio $\frac{{\mathbf{\varphi }}_{\mathbf{02}}}{{\mathbf{\varphi }}_{\mathbf{01}}}$of the magnetic flux through a turn in inductor 2 to that in inductor 1 is  . At time t=0 , the two switches are closed on B. At what time t is the flux through a turn in the two inductors equal?

At time t = 0, a $2.0 kg$particle has the position vector $\overrightarrow{r}= \left(4.0 m\right)\hat{i}-\left(2.0 m\right)\hat{j}$relative to the origin. Its velocity is given by$\overrightarrow{v=}\left(-6.0t2 m/s\right)$ $\hat{i}$for $t⩾ 0$  in seconds. About the origin, (a) What are the particle’s angular momentum $\overrightarrow{L}$ ? (b)What are the torque $\overrightarrow{\tau }$ acting on the particle, both in unit-vector notation and for $t> 0$ ?  (c) About the point $\left(-2.0 m, -3.0 m, 0\right),$what are $\overrightarrow{L }$? (d) About the point what are $\overrightarrow{\tau }$ for t 0?

A constant horizontal force moves a 50kg trunk 6.0 m up a $\mathbf{30}\mathbf{°}$ incline at constant speed. The coefficient of kinetic friction is 0.20. What are (a) the work done by the applied force and (b) the increase in the thermal energy of the trunk and incline?

Two waves,

$$\begin{gathered} {\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{50}\mathit{m}\mathit{m}\mathbf{)}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left[}\mathbf{(}\mathbf{25}\mathbf{.}\mathbf{1}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{/}\mathbf{m}\mathbf{)}\mathbf{x}\mathbf{-}\mathbf{(}\mathbf{440}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{t}\mathbf{\right]}\mathbf{ }\mathbf{and}\mathbf{ } \\ {\mathit{y}}_{\mathbf{2}}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{50}\mathit{m}\mathit{m}\mathbf{)}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left[}\mathbf{(}\mathbf{25}\mathbf{.}\mathbf{1}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{/}\mathbf{m}\mathbf{)}\mathbf{x}\mathbf{+}\mathbf{(}\mathbf{440}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{t}\mathbf{\right]} \end{gathered}$$

travel along a stretched string. (a) Plot the resultant wave as a function of t for, $\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathit{\lambda }\mathbf{/}\mathbf{8}\mathbf{,}\mathbf{ }\mathit{\lambda }\mathbf{/}\mathbf{4}\mathbf{,}\mathbf{ }\mathbf{3}\mathit{\lambda }\mathbf{/}\mathbf{8}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{\lambda }\mathbf{/}\mathbf{2}\mathbf{ }\mathbf{where}\mathbf{ }\mathit{\lambda }$ is the wavelength. The graphs should extend from t = 0 to a little over one period. (b) The resultant wave is the superposition of a standing wave and a traveling wave. In which direction does the traveling wave move? (c) How can you change the original waves so the resultant wave is the superposition of standing and traveling waves with the same amplitudes as before but with the traveling wave moving in the opposite direction? Next, use your graphs to find the place at which the oscillation amplitude is (d) maximum and (e) minimum. (f) How is the maximum amplitude related to the amplitudes of the original two waves? (g) How is the minimum amplitude related to the amplitudes of the original two waves?

In Fig. 13-57, identical blocks with identical masses m=2.00 kg hang from strings of different lengths on a balance at Earth’s surface.The strings have negligible mass and differ in length by h=5.00 cm. Assume Earth is spherical with a uniform density r=5.5. g/cm³. What is the difference in the weight of the blocks due to one being closer to Earth than the other?