Measurement

Can the sum of the magnitudes of two vectors ever be equal to the magnitude of the sum of the same two vectors? If no, why not? If yes, when?

The two vectors shown in Fig. 3-21 lie in a xy plane. What are the signs

of the x and y components, respectively, of (a)${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{1}}\mathbf{+}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{2}}\mathbf{,}\mathbf{ }\mathbf{\left(}\mathbf{b}\mathbf{\right)}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{1}}\mathbf{-}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{2}}\mathbf{,}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{ }{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{2}}\mathbf{-}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}}_{\mathbf{1}}$?

Question: An archer’s bow is drawn at its midpoint until the tension in the string is equal to the force exerted by the archer. What is the angle between the two halves of the string?

_{Figure 21-14 shows two charged particles on an axis. The charges are free to move. However, a third charged particle can be placed at a certain point such that all three particles are then in equilibrium. (a) Is that point to the left of the first two particles, to their right, or between them? (b) Should the third particle be positively or negatively charged? (c) Is the equilibrium stable or unstable?}

Figure 28-26 shows crossed uniform electric and magnetic fields $\underset{F}{\to }$and $\underset{B}{\to }$, at a certain instant, the velocity vectors of the 10 charged particles listed in Table 28-3. (The vectors are not drawn to scale.) The speeds given in the table are either less than or greater than  (see Question 5). Which particles will move out of the page toward you after the instant shown in Fig. 28-26?

In Fig 29-39 two circular arcs have radii $\mathbf{ }\mathbf{a}\mathbf{=}\mathbf{13}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{\text{cm}}\mathbf{ }\mathbf{\text{and}}\mathbf{ }\mathbf{b}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{7}\mathbf{ }\mathbf{\text{cm}}$, subtend angle $\mathbf{\theta }\mathbf{=}\mathbf{74}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{,}$ carry current $\mathit{i}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{411}\mathbf{ }\mathbf{\text{A}}$, and share the same center of curvature P. What are the (a)magnitude and  (b) direction (into or out of the page) of the net magnetic field at P?

(a) Rank the four paths of Fig $\mathbf{19}\mathbf{-}\mathbf{16}$according to the work done by the gas, greatest first. (b) Rank paths $\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{2}$ and $\mathbf{3}$ according to the change in the internal energy of the gas, most positive first and most negative last.

If $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{.}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{.}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}\mathbf{ }\mathbf{,}\mathbf{ }\mathit{m}\mathit{u}\mathit{s}\mathit{t}\mathbf{ }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{ }\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{l}\mathbf{ }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}$

An electron is in a state with I = 3 . (a) What multiple of  gives the magnitude of $\overrightarrow{\mathbf{L}}$? (b) What multiple of ${\mathbf{\mu }}_{\mathbf{B}}$ gives the magnitude of $\overrightarrow{\mathbf{\mu }}$ ? (c) What is the largest possible value of ${\mathbf{m}}_{\mathbf{1}}$ , (d) what multiple of $\mathit{ħ}$ gives the corresponding value of$\overrightarrow{\mathbf{L}}$ , and (e) what multiple of ${\mathbf{\mu }}_{\mathbf{B}}$ gives the corresponding value of  ${\mathbf{\mu }}_{\mathbf{orb}\mathbf{,}\mathbf{z}}$? (f) What is the value of the semi-classical angle $\mathbf{\theta }$ between the directions of ${\mathbf{L}}_{\mathbf{z}}\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{L}}$ ? What is the value of angle $\mathbf{\theta }$ for (g) the second largest possible value of  ${\mathbf{m}}_{\mathbf{I}}$ and (h) the smallest (that is, most negative) possible value of ${\mathbf{m}}_{\mathbf{I}}$ ?

An electron that is trapped in a one-dimensional infinite potential well of width $\mathbf{L}$is excited from the ground state to the first excited state. Does the excitation increase, decrease, or have no effect on the probability of detecting the electron in a small length of the x axis (a) at the center of the well and (b) near one of the well walls?

Suppose that an electron trapped in a one-dimensional infinite well of width $\mathbf{250}\mathit{p}\mathit{m}$  is excited from its first excited state to its third excited state. 

(a) What energy must be transferred to the electron for this quantum jump? The electron then de-excites back to its ground state by emitting light. In the various possible ways it can do this, what are the (b) shortest, (c) second shortest, (d) longest, and (e) second longest wavelengths that can be emitted? (f) Show the various possible ways on an energy-level diagram. If the light of wavelength   $\mathbf{29}\mathbf{.}\mathbf{4}\mathit{n}\mathit{m}$happens to be emitted, what are the (g) longest and (h) shortest wavelength that can be emitted afterward

A 100 W sodium lamp $\left(\lambda =589 \mathrm{nm}\right)$ radiates energy uniformly in all directions. (a) At what rate are photons emitted by the lamp? (b) At what distance from the lamp will a totally absorbing screen absorb photons at the rate of $1.00 \mathrm{photon}/{\mathrm{cm}}^2$ ? (c) What is the photon flux (photons per unit area per unit time) on a small screen 2.00 m from the lamp?

Figure 3-25 shows vector$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}$ and four other vectors that have the same magnitude but differ in orientation. (a) Which of those other four vectors have the same dot product with$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}$? (b) Which have a negative dot product with$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}$?

Question:  One clue used by your brain to determine the direction of a source of sound is the delay $∆t$ between the arrival of the sound at the ear closer to the source and the arrival at the father ear. Assume that the source is distant so that a wave front from it is approximately planar when it reaches you , and let D represent the separation between your ears.(a) If the source is located at angle  in front of you figure , what is $∆t$ in term of  D and the speed of sound  in air? (b) If you are submerged in water and the speed of sound is directly to your right, what is $∆t$ in terms of V_w and the speed of sound  in water? (c) Based on the time –delay clue, your brain interprets the submerged sound to arrive at an angle $\theta$ from the forward direction. Evaluate$\theta$  for fresh water at 20 ⁰C.

Question: In Figure, a constant horizontal force${\overrightarrow{\mathbf{F}}}_{\mathbf{a}\mathbf{p}\mathbf{p}}$ of magnitude is applied to a wheel of mass 10 kg   and radius0.30 m . The wheel rolls smoothly on the horizontal surface, and the acceleration of its centre of mass has magnitude 0.60 m/s²  (a) In unit-vector notation, what is the frictional force on the wheel? (b) What is the rotational inertia of the wheel about the rotation axis through its centre of mass? 

An ion source is producing ${}^{\mathbf{6}}\mathbf{Li}$ ions, which have charge $\mathbf{+}\mathbf{e}$ and mass $\mathbf{\text{9.99}}\mathbf{\times }{\mathbf{\text{10}}}^{\mathbf{-}\mathbf{27}}\mathbf{\text{kg}}$. The ions are accelerated by a potential difference of $\mathbf{\text{10kV}}$ and pass horizontally into a region in which there is a uniform vertical magnetic field of magnitude $\mathbf{B}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{\text{T}}$. Calculate the strength of the smallest electric field, to be set up over the same region that will allow the ${}^{\mathbf{6}}\mathbf{Li}$ ions to pass through un-deflected.

The x and y components of four vectors, $\overline{a}$,$\overrightarrow{b}$,$\overrightarrow{c}$  and $\overrightarrow{d}$are given below. For which vectors will your calculator give you the correct angle$\theta$ when you use it to find $\theta$ with Eq. 3-6? Answer first by examining Fig. 3-12, and then check your answers with your calculator.

$$\begin{gathered} a_x=3 a_y=3 c_x=-3 c_y=-3 \\ {b}_x=-3 b_y=3 d_x=3 d_y=-3 \end{gathered}$$

Question: Figure shows the output from a pressure monitor mounted at a point along the path taken by a sound wave of single frequency traveling at  343 m/s through air with a uniform density of 1.12 kg/m³ . The vertical axis scale is set by $\mathit{\Delta }{\mathbf{p}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{ }{\mathbf{p}}_{\mathbf{a}}$. If the displacement function of the wave is  $\mathbf{s}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{s}}_{\mathbf{m}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{kx}\mathbf{-}\mathbf{\omega t}\mathbf{)}\mathbf{,}$ what are (a) s_m (b) k (c) $\omega$ ? The air is then cooled so that its density is 1.35 kg/m³  and the speed of a sound wave through it is  320 m/s. The sound source again emits the sound wave at the same frequency and same pressure amplitude. Find the following quantities (d) s_m  (e) k  (f) $\omega$ ?

An 8.0 kg object is moving in the positive direction of an x axis. When it passes through $x = 0$, a constant force directed along the axis begins to act on it.

Figure 7-29 gives its kinetic energy K versus position x as it moves$x=0$ to $x=5.0$;$K_0= 30.0J$ . The force continues to act. What is v when the object moves back through $x = -3.0 m$?

What are the magnitudes of

 (a) the angular velocity, 

(b) the radial acceleration, and 

(c) the tangential acceleration of a spaceship taking a circular turn of radius $\mathbf{3220}\mathbf{ }\mathbf{\text{km}}$at a speed of $\mathbf{29}\mathbf{ }\mathbf{000}\raisebox{1ex}{${\displaystyle \mathbf{\text{km}}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{h}$}\right.$?

Question: Figure shoes four isotropic point sources of sound that are uniformly spaced on an x axis. The sources emit sound at the same wavelength $\mathit{\lambda }$ and same amplitude ${\mathit{s}}_{\mathbf{m}}$ and they emit in phase. A point, is shown on the y axis. Assume that as the sound waves travel to, the decrease in their amplitude is negligible. What multiple of  to P if distance d in the figure is $\mathbf{\left(}\mathbf{a}\mathbf{\right)}\frac{\mathbf{\lambda }}{\mathbf{4}}\mathbf{,}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\frac{\mathbf{\lambda }}{\mathbf{2}}\mathbf{,}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathit{\lambda }$

A pendulum consists of a$\text{2.0kg}$stone swinging on a string of $\text{4.0 m}$negligible mass. The stone has a speed of $\text{8.0m/s}$when it passes its lowest point.

1. What is the speed when the string is at${60}^0$to the vertical? 
2. What is the greatest angle with the vertical that the string will reach during the stone’s motion? 
3. If the potential energy of the pendulum–Earth system is taken to be zero at the stone’s lowest point, what is the total mechanical energy of the system?

The figure shows a pendulum of length $\mathrm{L}=1.25\mathrm{m}$. Its bob (which effectively has all the mass) has speed $v_0$ when the cord makes an angle  ${\mathrm{\theta }}_0=40.0°$with the vertical. 

1. What is the speed of the bob when it is in its lowest position if ${\mathrm{V}}_0=8.00\text{m/s}$? What is the least value that $v_0$ can have if the pendulum is to swing down and then up
2. To a horizontal position. 
3. To a vertical position with the cord remaining straight? 
4. Do the answers to (b) and (c) increase, decrease, or remain the same if they are increased by a few degrees?

A $60\mathrm{kg}$ skier starts from rest at height $\text{H=20m}$ above the end of a ski-jump ramp (Figure) and leaves the ramp at angle $\mathrm{\theta }=28°$. Neglect the effects of air resistance and assume the ramp is frictionless. 

1. What is the maximum height h of his jump above the end of the ramp? 
2. If he increased his weight by putting on a backpack, would h then be greater, less, or the same?

A block of mass $m\text{ = 2.0}$kg is dropped from height $\text{h=40 cm}$ onto a spring of spring constant $\text{k=1960 N/m}$ (Figure). Find the maximum distance the spring is compressed.

Two long, parallel copper wires of diameter 2.5 mm carry currents of 10A in opposite directions. (a) Assuming that their central axes are 20 mm apart, calculate the magnetic flux per meter of wire that exists in the space between those axes. (b) What percentage of this flux lies inside the wires? (c) Repeat part (a) for parallel currents.

At $t=0$  a $1.0 \text{Kg}$ ball is thrown from a tall tower with $\overrightarrow{\mathrm{v}}=\left(18 \mathrm{m}/\mathrm{s}\right)\hat{\mathrm{i}}+\left(24 \mathrm{m}/\mathrm{s}\right)\hat{\mathrm{j}}$ . What is ∆U of the ball-Earth system between $t=0$ and $t=6.0\text{s}$ (still free fall)?

In Fig. 22-50, a thin glass rod forms a semicircle of radius r=5.00 cm  . Charge is uniformly distributed along the rod, with +q=4.50pC   in the upper half and -q= -4.50pC  in the lower half. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electric field at P, the center of the semicircle?

In Fig 29-55, two long straight wires (shown in cross section) carry currents ${\mathbf{i}}_{\mathbf{1}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mA}$ and ${\mathbf{i}}_{\mathbf{1}}\mathbf{=}\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mA}$ directly out of the page. They are equal distances from the origin, where they set up a magnetic field. To what value must current be changed in order to rotate$\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{°}$ clockwise?

At one instant, force $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{F}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathit{N}$ acts on a 0.25kg object that has position vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{r}}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)}\mathit{m}$ and velocity vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)}\mathit{m}\mathbf{/}\mathit{s}$ About the origin and in unit-vector notation, (a) What is the object’s angular momentum? (b) What is the torque acting on the object?

$$\begin{gathered} \mathbf{A}\mathbf{ }\mathbf{particle}\mathbf{ }\mathbf{undergoes}\mathbf{ }\mathbf{uniform}\mathbf{ }\mathbf{circular}\mathbf{ }\mathbf{motion}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{radius}\mathbf{ }\mathbf{26}\mathbf{.}\mathbf{1}\mathbf{ }\mathbf{\mu }\mathbf{m}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{uniform}\mathbf{ }\mathbf{magnetic}\mathbf{ }\mathbf{field}\mathbf{.}\mathbf{ } \\ \mathbf{The}\mathbf{ }\mathbf{magnetic}\mathbf{ }\mathbf{force}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{particle}\mathbf{ }\mathbf{has}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{magnitude}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{60}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{17}}\mathbf{ }\mathbf{ }\mathbf{N}\mathbf{.} \\ \mathbf{ }\mathbf{What}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{kinetic}\mathbf{ }\mathbf{energy}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{particle}\mathbf{?} \end{gathered}$$

In the arrangement of Fig. $\mathbf{7}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{10}$, we gradually pull the block from $\mathit{x}\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{ }\mathit{t}\mathit{o}\mathbf{ }\mathbf{ }\mathit{x}\mathbf{=}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{c}\mathit{m}$ , where it is stationary. Figure  $7-35$gives the work that our force does on the block. The scale of the figure’s vertical axis is set by   ${\mathit{W}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{ }\mathit{J}$ . We then pull the block out to  $\mathit{x}\mathbf{=}\mathbf{+}\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{ }\mathit{c}\mathit{m}\mathbf{ }$ and release it from rest. How much work does the spring do on the block when the block moves from${\mathit{x}}_{\mathbf{i}}\mathbf{=}\mathbf{+}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{c}\mathit{m}$  to, (a) $\mathit{x}\mathbf{=}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{c}\mathit{m}$ (b) $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{ }\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{ }\mathit{c}\mathit{m}$, and (c) $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{ }\mathit{c}\mathit{m}$?

Calculate the height of the Coulomb barrier for the head-on collision of two deuterons, with effective radius 2.1 fm.

22-54, a non-conducting rod of length L=8.15cm   has a charge -q= -4.23fc  uniformly distributed along its length. (a) What is the linear charge density of the rod? What are the (b) magnitude and (c) direction (relative to the positive direction of the x axis) of the electric field produced at point P, at distance a=4.23fc  from the rod? What is the electric field magnitude produced at distance a=50m by (d) the rod and (e) a particle of charge -q = -4.23fC that we use to replace the rod? (At that distance, the rod “looks” likes a particle.)

Figure 7-37 gives spring force F_x versus position x for the spring–block arrangement of Fig . 7-10 . The scale is set by ${\mathbf{F}}_{\mathbf{x}}\mathbf{=}\mathbf{160}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{N}$. We release the block at $\mathbf{x}\mathbf{=}\mathbf{12}\mathbf{ }\mathbf{cm}$. How much work does the spring do on the block when the block moves from ${\mathbf{x}}_{\mathbf{i}}\mathbf{=}\mathbf{+}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$ to (a) x= +5.0 cm, (b) x=-5.0 cm, (c) x=-8.0 cm, and (d) x=-10.0 cm ?

The block in Fig. 7-10a  lies on a horizontal frictionless surface, and the spring constant is 50N/m. Initially, the spring is at its relaxed length and the block is stationary at position x=0. Then an applied force with a constant magnitude of  3.0 N pulls the block in the positive direction of the x axis, stretching the spring until the block stops. When that stopping point is reached, what are (a) the position of the block, (b) the work that has been done on the block by the applied force, and (c) the work that has been done on the block by the spring force? During the block’s displacement, what are (d) the block’s position when its kinetic energy is maximum and (e) the value of that maximum kinetic energy?

Question: At time t=0 , a 3.00 kg particle with velocity$\mathit{v}\mathbf{=}\left(5.0\text{m/s}\right)\widehat{\mathbf{\text{i}}}\mathbf{-}\left(6.0\text{m/s}\right)\widehat{\mathbf{\text{j}}}$ is at x =3.0 m and y= 8.0m. It pulled by a 7.0 N force in the negative x direction. About the origin, what are (a) the particle’s angular momentum, (b) the torque acting on the particle, and (c) the rate at which the angular momentum is changing?

Make a nuclidic chart similar to Fig. 42-6 for the 25 nuclides  ${\mathbf{ }}^{\mathbf{118}\mathbf{-}\mathbf{122}}\mathbf{Te}\mathbf{,}{\mathbf{ }}^{\mathbf{117}\mathbf{-}\mathbf{121}}\mathbf{Sb}\mathbf{,}{\mathbf{ }}^{\mathbf{116}\mathbf{-}\mathbf{120}}\mathbf{Sn}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{ }}^{\mathbf{115}\mathbf{-}\mathbf{119}}\mathbf{In}\mathbf{ }\mathbf{and}\mathbf{ }{\mathbf{ }}^{\mathbf{114}\mathbf{-}\mathbf{118}}\mathbf{Cd}$ . Draw in and label (a) all isobaric (constant A) lines and (b) all lines of constant neutron excess, defined as N - Z .

A 10 kg brick moves along an x axis. Its acceleration as a function of its position is shown in Fig.7-38. The scale of the figure’s vertical axis is set by as 20.0 m/s². What is the net work performed on the brick by the force causing the acceleration as the brick moves from $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$?

The force on a particle is directed along an x axis and given by $\mathbf{F}\mathbf{=}{\mathbf{F}}_{\mathbf{0}}\left(\frac{x}{x_0}-1\right)$. Find the work done by the force in moving the particle from $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{x}\mathbf{=}{\mathbf{x}}_{\mathbf{0}}$ by (a) plotting $\mathbf{F}\left(x\right)$ and measuring the work from the graph and (b) integrating $\mathbf{F}\left(x\right)$.

A boy is initially seated on the top of a hemispherical ice mound of radius R = 13.8 m. He begins to slide down the ice, with a negligible initial speed (Figure). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

A car moves along an x axis through a distance of 900 m, starting at rest (at $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}$) and ending at rest (at $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{900}\mathbf{ }\mathbf{m}$). Through the first $\frac{\mathbf{1}}{\mathbf{4}}$ of that distance, its acceleration is  $\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. Through the rest of that distance, its acceleration is  $\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{750}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. What are (a) its travel time through 900 m. (b) its maximum speed? (c) Graph position x, velocity v and acceleration a versus time t for the trip.

The radionuclide ${}^{\mathbf{64}}\mathbf{Cu}$ has a half-life of 12.7h. If a sample contains 5.50g of initially pure ${}^{\mathbf{64}}\mathbf{Cu}$at,  t = 0how much of it will decay between t = 14h and t = 16h?

Figure 10-35 shows three 0.0100kg particles that have been glued to a rod of length L=6.00cm and negligible mass. The assembly can rotate around a perpendicular axis through point O at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one?

In figure, ${\mathit{C}}_{\mathbf{1}}\mathbf{=}\mathbf{10}\mathbf{ }\mathit{\mu }\mathit{F}$ , ${\mathit{C}}_2\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathit{F}$ and ${\mathit{C}}_3\mathbf{=}\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathit{F}$ If no capacitor can withstand a potential difference of more than 100 V without failure, (a)What is the magnitude of the maximum potential difference that can exist between points A and B? (b)What is the maximum energy that can be stored in the three capacitor arrangement?

Cars A and B move in the same direction in adjacent lanes. The position x of car A is given in Fig. 2-30, from time $\mathit{t}\mathbf{=}\mathbf{0}$  to time  $\mathit{t}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$. The figure’s vertical scaling is set by 
${\mathit{x}}_{\mathbf{s}}\mathbf{=}\mathbf{32}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$. At $\mathit{t}\mathbf{=}\mathbf{0}$, car B is at $\mathit{x}\mathbf{=}\mathbf{0}$ , with a velocity of $\mathbf{12}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$ and a negative constant acceleration  ${\mathit{a}}_{\mathbf{B}}$. (a) What must ${\mathit{a}}_{\mathbf{B}}$ be such that the cars are (momentarily) side by side (momentarily at the same value of x) at  $\mathit{t}\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$? (b)For that value of ${\mathit{a}}_{\mathbf{B}}$_, how many times are the cars side by side? (c) Sketch the position x  of car B vs time t  on Fig. 2-30.How many times will the cars be side by side if the magnitude of acceleration ${\mathit{a}}_{\mathbf{B}}$  is (d) more than and (e) less than answer to part a.?

A can of sardines is made to move along an x axis from $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}$ to $\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{25}\mathbf{ }\mathbf{m}$ by a force with a magnitude given by $\mathbf{F}\mathbf{=}\mathbf{exp}\left(-4x^2\right)$ with x in meters and F in newtons. (Here exp is the exponential function.) How much work is done on the can by the force?

A projectile is shot directly away from Earth’s surface.Neglect the rotation of Earth. What multiple of Earth’s radius  gives the radial distance a projectile reaches if (a) its initial speed is  $\mathbf{0}\mathbf{.}\mathbf{500}$ of the escape speed from Earth and(b) its initial kinetic energy is $\mathbf{0}\mathbf{.}\mathbf{500}$ of the kinetic energy required to escape Earth?(c)What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

You are driving towards a traffic signal when it turns yellow. Your speed limit is the legal speed limit of  ${\mathit{v}}_{\mathbf{0}}\mathbf{=}\mathbf{55}\mathbf{ }\mathbf{km}\mathbf{/}\mathbf{h}$; your best deceleration rate has a magnitude  $\mathit{a}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{18}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. Your best reaction time to begin braking is  $\mathit{T}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{.}\mathbf{75}\mathbf{ }\mathit{s}$. To avoid having the front of your car enter the intersection after the light turns red, should you brake to a stop or continue to move at $\mathbf{55}\mathbf{ }\mathbf{ }\mathbf{km}\mathbf{/}\mathbf{h}$  if the distance to the intersection and the duration of yellow light are (a) $\mathbf{40}\mathbf{ }\mathit{m}$  and $\mathbf{2}\mathbf{.}\mathbf{8}\mathbf{ }\mathit{s}$ (b) $\mathbf{32}\mathbf{ }\mathit{m}$  and  $\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{ }\mathit{s}$.Give an answer of brake, continue, either (if either strategy works) or neither (if neither strategy works and yellow duration is inappropriate).

Figure 36-45 gives the parameter $\mathit{\beta }$ of Eq. 36-20 versus the sine of the angle in a two-slit interference experiment using light of wavelength 435 nm. The vertical axis scale is set by ${\mathit{\beta }}_{\mathbf{s}}\mathbf{=}\mathbf{80}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{rad}}$. What are (a) the slit separation, (b) the total number of interference maxima (count them on both sides of the pattern’s center), (c) the smallest angle for a maxima, and (d) the greatest angle for a minimum? Assume that none of the interference maxima are completely eliminated by a diffraction minimum.