Classical Mechanics

John R. Taylor

Chapter 9 Mechanics in Noninertial Frames - all with Video Answers

Educators

Chapter Questions

08:16

Problem 1

Be sure you understand why a pendulum in equilibrium hanging in a car that is accelerating forward tilts backward, and then consider the following: A helium balloon is anchored by a massless string to the floor of a car that is accelerating forward with acceleration $A$. Explain clearly why the balloon tends to tilt forward and find its angle of tilt in equilibrium. [Hint: Helium balloons float because of the buoyant Archimedean force, which results from a pressure gradient in the air. What is the relation between the directions of the gravitational field and the buoyant force?]

Jacob Schulze

Numerade Educator

01:47

Problem 2

A donut-shaped space station (outer radius $R$ ) arranges for artificial gravity by spinning on the axis of the donut with angular velocity $\omega .$ Sketch the forces on, and accelerations of, an astronaut standing in the station (a) as seen from an inertial frame outside the station and (b) as seen in the astronaut's personal rest frame (which has a centripetal acceleration $A=\omega^{2} R$ as seen in the inertial frame). What angular velocity is needed if $R=40$ meters and the apparent gravity is to equal the usual value of about $10 \mathrm{m} / \mathrm{s}^{2} ?$ (c) What is the percentage difference between the perceived $g$ at a six-foot astronaut's feet $(R=40 \mathrm{m})$ and at his head $(R=38 \mathrm{m}) ?$

Victor Salazar

Numerade Educator

03:56

Problem 3

(a) Consider the tidal force (9.12) on a mass $m$ at the position $P$ of Figure 9.4. Write $d$ as $\left(d_{\mathrm{o}}-R_{\mathrm{e}}\right)=d_{\mathrm{o}}\left(1-R_{\mathrm{e}} / d_{\mathrm{o}}\right)$ and use the binomial approximation $(1-\epsilon)^{-2} \approx 1+2 \epsilon$ to show that $\mathbf{F}_{\mathrm{tid}} \approx-\left(2 G M_{\mathrm{m}} m R_{\mathrm{e}} / d_{\mathrm{o}}^{3}\right) \hat{\mathbf{x}} .$ Confirm the direction of the force shown in Figure 9.4 and make a numerical comparison of the tidal force with the gravitational force $m \mathbf{g}$ of the earth. (b) Do the corresponding calculations for the force at the point $R$. Compare this force with that of part (a) (magnitude and direction).

Andy Chen

Numerade Educator

09:52

Problem 4

Do the same calculations as in Problem 9.3(a) but for the tidal force at the point $Q$ in Figure 9.4. [In this case write $\left.\hat{\mathbf{d}} / d^{2}=\mathbf{d} / d^{3} \text { and use the binomial approximation in the form }(1+\epsilon)^{-3} \approx 1-3 \epsilon .\right]$

Donald Albin

Numerade Educator

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Problem 5

Review the derivation of the tidal potential energy (9.16) of a drop of water at the point $Q$ in Figure 9.5 and then give in detail the derivation of (9.17) for the tidal $\mathrm{PE}$ at the point $P$.

Victor Salazar

Numerade Educator

01:29

Problem 6

Let $h(\theta)$ denote the height of the ocean at any point $T$ on the surface, where $h(\theta)$ is measured up from the level at the point $Q$ of Figure 9.5 and $\theta$ is the polar angle $T O R$ of $T$. Given that the surface of the ocean is an equipotential, show that $h(\theta)=h_{\mathrm{o}} \cos ^{2} \theta,$ where $h_{\mathrm{o}}=3 M_{\mathrm{m}} R_{\mathrm{e}}^{4} /\left(2 M_{\mathrm{e}} d_{\mathrm{o}}^{3}\right) .$ Sketch and describe the shape of the ocean's surface, bearing in mind that $h_{\mathrm{o}} \ll R_{\mathrm{e}}$. [Hint: You will need to evaluate $U_{\mathrm{tid}}(T)$ as given by $(9.13),$ with $d$ equal to the distance $M T .$ To do this you need to find $d$ by the law of cosines and then approximate $d^{-1}$ using the binomial approximation, being very careful to keep all terms through order $\left(R_{\mathrm{e}} / d_{\mathrm{o}}\right)^{2} .$ Neglect any effects of the sun.]

Narayan Hari

Numerade Educator

03:45

Problem 7

(a) Explain the relation (9.30) between the derivatives of a vector $\mathbf{Q}$ in two frames $\boldsymbol{\delta}_{\mathbf{o}}$ and $\boldsymbol{\delta}$ for the special case that $Q$ is fixed in the frame $S$. (b) Do the same for a vector $Q$ that is fixed in the frame $\delta_{0}$ and compare with your answer to part (a).

Susan Hallstrom

Numerade Educator

05:30

Problem 8

What are the directions of the centrifugal and Coriolis forces on a person moving (a) south near the North Pole, (b) east on the equator, and (c) south across the equator?

Jake Rempel

Numerade Educator

09:59

Problem 9

A bullet of mass $m$ is fired with muzzle speed $v_{\mathrm{o}}$ horizontally and due north from a position at colatitude $\theta$. Find the direction and magnitude of the Coriolis force in terms of $m, v_{\mathrm{o}}, \theta,$ and the earth's angular velocity $\Omega .$ How does the Coriolis force compare with the bullet's weight if $v_{\mathrm{o}}=1000 \mathrm{m} / \mathrm{s}$ and $\theta=40$ deg?

Coleen Amado

Numerade Educator

10:07

Problem 10

The derivation of the equation of motion (9.34) for a rotating frame made the assumption that the angular velocity $\boldsymbol{\Omega}$ was constant. Show that if $\dot{\boldsymbol{\Omega}} \neq 0$ then there is a third "fictitious force," sometimes called the azimuthal force, on the right side of (9.34) equal to $m \mathbf{r} \times \dot{\mathbf{\Omega}}$.

Sarah Mccrumb

Numerade Educator

05:10

Problem 11

In this problem you will prove the equation of motion (9.34) for a rotating frame using the Lagrangian approach. As usual, the Lagrangian method is in many ways easier than the Newtonian (except that it calls for some slightly tricky vector gymnastics), but is perhaps less insightful. Let $\delta$ be a noninertial frame rotating with constant angular velocity $\boldsymbol{\Omega}$ relative to the inertial frame $\boldsymbol{\delta}_{\mathrm{o}}$. Let both frames have the same origin, $O=O^{\prime} .$ (a) Find the Lagrangian $\mathcal{L}=T-U$ in terms of the coordinates r and $\dot{\mathbf{r}}$ of $\delta$. [Remember that you must first evaluate $T$ in the inertial frame. In this connection, recall that $\mathbf{v}_{\mathbf{o}}=\mathbf{v}+\mathbf{\Omega} \times \mathbf{r} . \mathbf{J}(\mathbf{b})$ Show that the three Lagrange equations reproduce (9.34) precisely.

Keshav Singh

Numerade Educator

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Problem 12

(a) Show that to design a static structure in a rotating frame (such as a space station) one can use the ordinary rules of statics except that one must include the extra "fictitious" centrifugal force.
(b) I wish to place a puck on a rotating horizontal turntable (angular velocity $\Omega$ ) and to have it remain at rest on the table, held by the force of static friction (coefficient $\mu$ ). What is the maximum distance from the axis of rotation at which I can do this? (Argue from the point of view of an observer in the rotating frame.)

Lainey Roebuck

Numerade Educator

13:43

Problem 13

Show that the angle $\alpha$ between a plumb line and the direction of the earth's center is well approximated by $\tan \alpha=\left(R_{\mathrm{e}} \Omega^{2} \sin 2 \theta\right) /(2 g),$ where $g$ is the observed free-fall acceleration and we assume the earth is perfectly spherically symmetric. Estimate the maximum and minimum values of the magnitude of $\alpha$.

Linda Winkler

Numerade Educator

01:03

Problem 14

I am spinning a bucket of water about its vertical axis with angular velocity $\Omega$. Show that, once the water has settled in equilibrium (relative to the bucket), its surface will be a parabola. (Use cylindrical polar coordinates and remember that the surface is an equipotential under the combined effects of the gravitational and centrifugal forces.)

Elizabeth Vilchock

Numerade Educator

01:11

Problem 15

On a certain planet, which is perfectly spherically symmetric, the free-fall acceleration has magnitude $g=g_{0}$ at the North Pole and $g=\lambda g_{0}$ at the equator (with $0 \leq \lambda \leq 1$ ). Find $g(\theta)$, the freefall acceleration at colatitude $\theta$ as a function of $\theta$.

Carson Merrill

Numerade Educator

03:00

Problem 16

The center of a long frictionless rod is pivoted at the origin and the rod is forced to rotate at a constant angular velocity $\Omega$ in a horizontal plane. Write down the equation of motion for a bead that is threaded on the rod, using the coordinates $x$ and $y$ of a frame that rotates with the rod (with $x$ along the rod and $y$ perpendicular to it). Solve for $x(t)$. What is the role of the centrifugal force? What of the Coriolis force?

Prabhat Tyagi

Numerade Educator

02:37

Problem 17

Consider the bead threaded on a circular hoop of Example 7.6 (page 260 ), working in a frame that rotates with the hoop. Find the equation of motion of the bead, and check that your result agrees with Equation (7.69). Using a free-body diagram, explain the result (7.71) for the equilibrium positions.

Ryan Hood

Numerade Educator

01:54

Problem 18

A particle of mass $m$ is confined to move, without friction, in a vertical plane, with axes $x$ horizontal and $y$ vertically up. The plane is forced to rotate with constant angular velocity $\Omega$ about the $y$ axis. Find the equations of motion for $x$ and $y,$ solve them, and describe the possible motions.

Katie Mcalpine

Numerade Educator

02:43

Problem 19

I am standing (wearing crampons) on a perfectly frictionless flat merry-go-round, which is rotating counterclockwise with angular velocity $\Omega$ about its vertical axis. (a) I am holding a puck at rest just above the floor (of the merry-go-round) and release it. Describe the puck's path as seen from above by an observer who is looking down from a nearby tower (fixed to the ground) and also as seen by me on the merry-go-round. In the latter case explain what I see in terms of the centrifugal and Coriolis forces. (b) Answer the same questions for a puck which is released from rest by a long-armed spectator who is standing on the ground leaning over the merry-go-round.

Nick Johnson

Numerade Educator

01:49

Problem 20

Consider a frictionless puck on a horizontal turntable that is rotating counterclockwise with angular velocity $\Omega$. (a) Write down Newton's second law for the coordinates $x$ and $y$ of the puck as seen by me standing on the turntable. (Be sure to include the centrifugal and Coriolis forces, but ignore the earth's rotation.) (b) Solve the two equations by the trick of writing $\eta=x+$ iy and guessing a solution of the form $\eta=e^{-i \alpha t} .$ [In this case $-$ as in the case of critically damped SHM discussed in Section $5.4-$ you get only one solution this way. The other has the same form (5.43) we found for the second solution in damped SHM.] Write down the general solution. (c) At time $t=0,$ I push the puck from position $\mathbf{r}_{\mathrm{o}}=\left(x_{\mathrm{o}}, 0\right)$ with velocity $\mathbf{v}_{\mathrm{o}}=\left(v_{\mathrm{xo}}, v_{\mathrm{yo}}\right)($ all as measured by me on the turntable). Show that
$$\left.\begin{array}{l}
x(t)=\left(x_{0}+v_{x} t\right) \cos \Omega t+\left(v_{y o}+\Omega x_{o}\right) t \sin \Omega t \\
y(t)=-\left(x_{0}+v_{x 0} t\right) \sin \Omega t+\left(v_{y o}+\Omega x_{o}\right) t \cos \Omega t
\end{array}\right\}.$$
(d) Describe and sketch the behavior of the puck for large values of $t .$ [Hint: When $t$ is large the terms proportional to $t$ dominate (except in the case that both their coefficients are zero). With $t$ large, write
(9.72) in the form $x(t)=t\left(B_{1} \cos \Omega t+B_{2} \sin \Omega t\right),$ with a similar expression for $y(t),$ and use the trick
of (5.11) to combine the sine and cosine into a single cosine $-$ or sine, in the case of $y(t) .$ By now you can recognize that the path is the same kind of spiral, whatever the initial conditions (with the one exception mentioned).]

Manish Jain

Numerade Educator

02:43

Problem 21

When a puck slides on a rotating turntable, as in Problems 9.20 and 9.24, it can come instantaneously to rest. Sketch the shape of the path when this happens and explain. If you did Problem 9.24, comment on the relevance of this result to part (d) of that problem.

Nick Johnson

Numerade Educator

02:37

Problem 22

If a negative charge $-q$ (an electron, for example) in an elliptical orbit around a fixed positive charge $Q$ is subjected to a weak uniform magnetic field $\mathbf{B}$, the effect of $\mathbf{B}$ is to make the ellipse precess slowly - an effect known as Larmor precession. To prove this, write down the equation of motion of the negative charge in the field of $Q$ and $\mathbf{B}$. Now rewrite it for a frame rotating with angular velocity $\boldsymbol{\Omega}$. [Remember that this changes both $d^{2} \mathbf{r} / d t^{2}$ and $d \mathbf{r} / d t .$ ] Show that by suitable choice of
S you can arrange that the terms involving $\dot{\mathbf{r}}$ cancel out, but that you are left with one term involving
$\mathbf{B} \times(\mathbf{B} \times \mathbf{r}) .$ If $\mathbf{B}$ is weak enough this term can certainly be neglected. Show that in this case the orbit
in the rotating frame is an ellipse (or hyperbola). Describe the appearance of the ellipse as seen in the original nonrotating frame.

Narayan Hari

Numerade Educator

10:34

Problem 23

Here is an unusual way to solve the two-dimensional isotropic oscillator - the motion of a particle subject to a force - $k \mathbf{r}$. Show that by choosing a suitable rotating reference frame, you can arrange that the centrifugal force exactly cancels the force $-k \mathbf{r}$. Recalling the analogy between the Coriolis and magnetic forces, you should be able to write down the general solution for the motion as seen in the rotating frame. If you write your solution in the complex form of Section $2.7,$ then you can transform back to the nonrotating frame by multiplying by a suitable rotating complex number. Show that the general solution is an ellipse. [See Problem 8.11 for some guidance on this last part.]

Guilherme Barros

Numerade Educator

02:43

Problem 24

[Computer] Use a suitable plotting program (such as ParametricPlot in Mathematica) to plot the orbits (9.72) of the puck of Problem 9.20 on a rotating turntable with $x_{\mathrm{o}}=\Omega=1$ and the following initial velocities $\mathbf{v}_{\mathbf{o}}:(\mathbf{a})(0,1),(\mathbf{b})(0,0),(\mathbf{c})(0,-1),(\mathbf{d})(-0.5,-0.5),(\mathbf{e})(-0.7,-0.7),$
(f) (0,-0.1) . Comment on any interesting features.

Nick Johnson

Numerade Educator

01:03

Problem 25

A high-speed train is traveling at a constant 150 m/s (about 300 mph) on a straight, horizontal track across the South Pole. Find the angle between a plumb line suspended from the ceiling inside the train and another inside a hut on the ground. In what direction is the plumb line on the train deflected?

Zulfiqar Ali

Numerade Educator

11:14

Problem 26

In Section 9.8, we used a method of successive aproximations to find the orbit of an object that is dropped from rest, correct to first order in the earth's angular velocity $\Omega$. Show in the same way that if an object is thrown with initial velocity $\mathbf{v}_{\mathrm{o}}$ from a point $O$ on the earth's surface at colatitude $\theta$. then to first order in $\Omega$ its orbit is
$$\left.\begin{array}{l}

x=v_{x_{0} t}+\Omega\left(v_{y_{0}} \cos \theta-v_{z 0} \sin \theta\right) t^{2}+\frac{1}{3} \Omega g t^{3} \sin \theta \\
y=v_{y 0} t-\Omega\left(v_{x 0} \cos \theta\right) t^{2} \\
z=v_{z 0} t-\frac{1}{2} g t^{2}+\Omega\left(v_{x 0} \sin \theta\right) t^{2}.
\end{array}\right\}$$
[First solve the equations of motion (9.53) in zeroth order, that is, ignoring $\Omega$ entirely. Substitute your zeroth-order solution for $\dot{x}, \dot{y},$ and $\dot{z}$ into the right side of equations (9.53) and integrate to give the next approximation. Assume that $v_{\mathrm{o}}$ is small enough that air resistance is negligible and that $\mathrm{g}$ is a constant throughout the flight.]

Khoobchandra Agrawal

Numerade Educator

06:19

Problem 27

In Section 9.8, we discussed the path of an object that is dropped from a very tall stepladder above the equator. (a) Sketch this path as seen from a tower to the north of the drop and fixed to the earth. Explain why the object lands to the east of its point of release. (b) Sketch the same experiment as seen by an inertial observer floating in space to the north of the drop. Explain clearly (from this point of view) why the object lands to the east of its point of release. [Hint: The object's angular momentum about the earth's center is conserved. This means that the object's angular velocity $\dot{\phi}$ changes as it falls.]

Guilherme Barros

Numerade Educator

05:52

Problem 28

Use the result (9.73) of Problem 9.26 to do the following: A naval gun shoots a shell at colatitude $\theta$ in a direction that is $\alpha$ above the horizontal and due east, with muzzle speed $v_{\mathrm{o}}$. (a) Ignoring the earth's rotation (and air resistance), find how long $(t)$ the shell would be in the air and how far away
( $R$ ) it would land. If $v_{\mathrm{o}}=500 \mathrm{m} / \mathrm{s}$ and $\alpha=20^{\circ},$ what are $t$ and $R ?$ (b) A naval gunner spots an enemy ship due east at the range $R$ of part (a) and, forgetting about the Coriolis effect, aims his gun exactly as in part (a). Find by how far north or south, and in which direction, the shell will miss the target, in terms of $\Omega, v_{\mathrm{o}}, \alpha, \theta,$ and $g .$ (It will also miss in the east-west direction but this is perhaps less critical.)
If the incident occurs at latitude $50^{\circ}$ north $\left(\theta=40^{\circ}\right),$ what is this distance? What if the latitude is $50^{\circ}$ south? This problem is a serious issue in long-range gunnery: In a battle near the Falkland Islands in World War I, the British navy consistently missed German ships by many tens of yards because they apparently forgot that the Coriolis effect in the southern hemisphere is opposite to that in the north.

Emily Anderson

Numerade Educator

11:10

Problem 29

(a) A baseball is thrown vertically up with initial speed $v_{\mathrm{o}}$ from a point on the ground at colatitude $\theta .$ Use the solution (9.73) of Problem 9.26 to show that the ball will return to the ground a distance $\left(4 \Omega v_{\mathrm{o}}^{3} \sin \theta\right) /\left(3 g^{2}\right)$ to the west of its launch point. (b) Estimate the size of this effect on the equator if $v_{\mathrm{o}}=40 \mathrm{m} / \mathrm{s} .$ (c) Sketch the ball's orbit as seen from the north (by an observer fixed to the earth). Compare with the orbit of a ball dropped from a point above the equator, and explain why the Coriolis effect moves the dropped ball to the east, but the thrown ball to the west.

Susan Hallstrom

Numerade Educator

03:17

Problem 30

The Coriolis force can produce a torque on a spinning object. To illustrate this, consider a horizontal hoop of mass $m$ and radius $r$ spinning with angular velocity $\omega$ about its vertical axis at colatitude $\theta$. Show that the Coriolis force due to the earth's rotation produces a torque of magnitude $m \omega \Omega r^{2} \sin \theta$ directed to the west, where $\Omega$ is the earth's angular velocity. This torque is the basis of the gyrocompass.

Khoobchandra Agrawal

Numerade Educator

04:52

Problem 31

The Compton generator is a beautiful demonstration of the Coriolis force due to the earth's rotation, invented by the American physicist A. H. Compton (1892-1962, best known as author of the Compton effect) while he was still an undergraduate. A narrow glass tube in the shape of a torus or ring (radius $R$ of the ring $\gg$ radius of the tube) is filled with water, plus some dust particles to let one see any motion of the water. The ring and water are initially stationary and horizontal, but the ring is then spun through $180^{\circ}$ about its east-west diameter. Explain why this should cause the water to move around the tube. Show that the speed of the water just after the $180^{\circ}$ turn should be $2 \Omega R \cos \theta,$ where $\Omega$ is the earth's angular velocity, and $\theta$ is the colatitude of the experiment. What would this speed be if $R \approx 1 \mathrm{m}$ and $\theta=40^{\circ} ?$ Compton measured this speed with a microscope and got agreement within 3\%.

Khoobchandra Agrawal

Numerade Educator

01:16

Problem 32

Do all parts of Problem 9.28, but find the distance by which the shell misses its target in both the north-south and east-west directions. [Hint: In this case you must recognize that the time of fight is affected by the Coriolis effect.]

Yuou Sun

Numerade Educator

01:14

Problem 33

The general solution for the small-amplitude motion of a Foucault pendulum is given by (9.66). If at $t=0$ the pendulum is at rest with $x=A$ and $y=0$, find the two coefficients $C_{1}$ and $C_{2}$, and show that because $\Omega \ll \omega_{\mathrm{o}}$ they are well approximated as $C_{1}=C_{2}=A / 2,$ giving the solution (9.67).

Raj Bala

Numerade Educator

07:43

Problem 34

v9.34 \star\star\star At a point $P$ on the earth's surface, an enormous perfectly flat and frictionless platform is built. The platform is exactly horizontal $-$ that is, perpendicular to the local free-fall acceleration $\mathbf{g}_{P}$. Find the equation of motion for a puck sliding on the platform and show that it has the same form as
(9.61) for the Foucault pendulum except that the pendulum's length $L$ is replaced by the earth's radius
R. What is the frequency of the puck's oscillations and what is that of its Foucault precession? [Hints: Write the puck's position vector, relative to the earth's center $O$ as $\mathbf{R}+\mathbf{r},$ where $\mathbf{R}$ is the position of the point $P$ and $\mathbf{r}=(x, y, 0)$ is the puck's position relative to $P .$ The contribution to the centrifugal force involving $\mathbf{R}$ can be absorbed into $\mathbf{g}_{P}$ and the contribution involving $\mathbf{r}$ is negligible. The restoring force comes from the variation of $g$ as the puck moves. $J$ To check the validity of your approximations, compare the approximate size of the gravitational restoring force, the Coriolis force, and the neglected term $m(\boldsymbol{\Omega} \times \mathbf{r}) \times \boldsymbol{\Omega}$ in the centrifugal force.

Katie Mcalpine

Numerade Educator