Mechanics Berkeley Physics

Charles Kittel, Walter D. Knight, Malvin A. Ruderman, A. Carl Helmholz, Burton J. Moyer

Chapter 6

Conservation of Linear and Angular Momentum - all with Video Answers

Educators

AP

Chapter Questions

Problem 1

Angular momentum of a satellite
(a) What is the angular momentum (referred to the center of the orbit) of a satellite of mass $M_{\mathrm{s}}$ that moves in a circular orbit of radius $r ?$ The result is to be expressed in terms only of $r, G, M_{e}, M_{e}$ (the mass of the earth).
(b) For $M_{n}=100 \mathrm{~kg}$, what is the numerical value (in egs units of the angular momentum of an orbit for which the radius is twice the radius of the earth?

Ajay Singhal

Ajay Singhal

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Problem 2

Frictional effects on satellite motion
(a) What is the effect of atmospheric friction on the motion of a satellite in a circular (or nearly circular) orbit? Why does friction inerease the satellite velocity?
(b) Does friction increase or decrease the angular momentum of the satellite measured with respect to the center of the earth? Why?

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Problem 3

Energy-angular momentum relation for a satellite. Express in terms of the angular momentum $J$ the kinetic, potential, and total energy of a satellite of mass $M$ in a circular orbit of radius $r$.

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Problem 4

Electron bound to a proton. An electron moves about a proton in a circular orbit of radius $0.5 \AA \equiv 0.5 \times 10^{-8} \mathrm{~cm}$.
(a) What is the orbital angular momentum of the electron about the proton?
(b) What is the total energy (expressed in ergs and in electron volts)?
(c) What is the ionization energy, that is, the energy that must be given to the electron to separate it from the proton?

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Problem 5

Internal torques stum to zero. Consider the isolated system of three particles 1,2, and 3 (shown in Fig. $6.23$ ) interacting with central forces $F_{12}=1 \mathrm{dyn}, F_{13}=0.6 \mathrm{dyn}$, and $F_{23}=$ $0.75 \mathrm{dyn}$, where $F_{\mathrm{i} /}$ denotes the force on particle $i$ when it interacts with particle $/ .$ Pick two different points and show that for each the sum of the torques about that point is zero.

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Problem 6

Forces on ladtler. A ladder of mass $20 \mathrm{~kg}$ and length $10 \mathrm{~m}$ rests against a slippery vertical wall at an angle of $30^{\circ}$ with the vertical. The ladder, of uniform construction, is prevented from slipping by friction with the ground. What is the magnitude in dynes of the force exerted by the ladder on the wall? (Hint: Use the fact that the torques must sum to zero for a $\begin{array}{ll}\text { ladder at rest.)

Ajay Singhal

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Problem 7

Kinetic energy of center of mass. In a collision of a particle of mass $m_{1}$ moving with speed $v_{1}$ with a stationary particle of mass $m_{2}$ not all the original kinetic energy can be converted into heat or internal energy. What fraction ean be so converted? Show that this energy is just that equal to the kinetic energy in the center-of-mass system.

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Problem 8

Falling chain. A chain of mass $M$ and length $l$ is coiled up on the edge of a table. A very small length at one end is pushed off the edge and starts to fall under the force of gravity, pulling more and more of the chain off the table. Assume that the velocity of each element remains zero until it is jerked into motion with the velocity of the falling section. Find the velocity when a length $x$ has fallen off.
When the entire length $l$ is just off the table, what fraction of the original potential energy has been converted into the kinetic energy of translation of the chain?

Surendra Kumar

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Problem 9

Angular momentum in near collision of two particles. A neutron of energy $1 \mathrm{MeV}$ passes a proton at such a distance that the angular momentum of the neutron relative to the proton approximately equals $10^{-26}$ erg-s. What is the distance of closest approach? (We neglect the energy of interaction between the two particles.)

Ajay Singhal

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Problem 10

Coeffictent of restitution. The coefficient of restitution $r$ is defined for two bodies as the velocity of separation divided by the velocity of approach $0 \leq r \leq 1$. It can be used in collision problems to provide the solution that otherwise could be provided by an energy relation. (a) Show that if the coefficient of restitution between a ball and a massive flat horizontal plate is $r$, the height to which the ball will rise after $n$ bounces is $h_{1} r^{2 n}$, where $h_{0}$ is the height from which it was dropped.
(b) Show that for a head-on collision of two bodies of masses $m_{1}$ and $m_{2}$ with coefficient $r$, the loss in kinetic energy is $\left(1-r^{2}\right)$ times the kinetic energy in the center-of-mass system.

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Problem 11

Particle-dumbbell collision. Two equal masses $M$ are connected by a rigid rod of negligible mass and of length $a$. The center of mass of this dumbbell-like system is stationary in gravity-free space, and the system rotates about the center of mass with angular velocity $\omega .$ One of the rotating masses strikes head-on a third stationary mass $M$, and the two stick together.
(a) Locate the center of mass of the three-particle system at the instant prior to collision. What is the velocity of the center of mass? (Note: This is not the velocity of the point on the rigid rod that instantaneously coincides with the center of mass.)
(b) What is the angular momentum of the three-mass system about the center of mass at the instant prior to collision? At the instant following collision?
(c) What is the angular velocity of the system about the center of mass after the collision?
(d) What are the initial and final kinetie energies?

AP

Andreas Papavassiliou

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Problem 12

Angular momentum of tetherball. The object of the game tetherball (Fig. 6.24) is to hit the ball hard enough and fast enough to wind its tether cord in one direction about the vertical post to which it is tied before the opposing player can wind it in the opposite direction. The game is exciting, and the dynamics of the ball's motion are complicated. Let us examine a simple type of motion in which the ball moves in a horizontal plane in a spiral of decreasing radius as the cord winds round the post after a single blow that gives the ball an initial speed $v_{i 0}$. The length of the cord is $l$ and the radius of the post is $a \ll l$.
(a) What is the instantaneous center of revolution?
(b) Is there a torque about the axis through the center of the post? Is angular momentum conserved?
(c) Assume that kinetic energy is conserved and calculate the speed as a function of time.
(d) What is the angular velocity after the ball has made five complete revolutions?

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Problem 13

Effective centrifugal potential energy. It is convenient to use plane polar coordinates $r, \theta$ for motion in a plane perpendicular to an axis of rotation
(a) Show that the velocity in such a coordinate system may be written
$$
\mathbf{v}=v_{r} \hat{\mathbf{r}}+\mathbf{v}_{e} \dot{\boldsymbol{\theta}}
$$
where $v_{r}$ is just $d r / d t$, the rate of change of the length $r$, and
$$
v_{\theta}=r \frac{d \theta}{d t}
$$
(b) Show that the kinetic energy of a particle in this coordinate system is
$$
K=\frac{1}{2} M\left(\dot{r}^{2}+\omega^{2} r^{2}\right)
$$
where $\omega=d \theta / d t$
(c) Show that the total energy is
$$
E=U(r)+\frac{1}{2} M r^{2}+\frac{J^{2}}{2 M r^{2}}
$$
where $J$ is the angular momentum of the particle about the fixed axis normal to the plane of the motion. Hint:
Recall Eq. (6.37).
(d) Because the force is central there is no torque on the particle and $J$ is a constant of the motion. The term $J^{2} / 2 M r^{2}$ is sometimes called the centrifugal potential energy. Show that the centrifugal potential energy represents an outward radial force $J^{2} / M r^{3}$.
(e) If $U(r)=12 \mathrm{Cr}^{2}$, show that $U(r)$ represents an inward radial force $-\mathrm{Cr}$ Show from $(d)$ and $(e)$ that the balance of these forces is equivalent to the condition $\omega^{2}=C / M$.

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Problem 14

Rocket in earth's field. A rocket of initial mass $M_{0}$ burns an adjustable amount of fuel $\beta \mathrm{g} / \mathrm{s} .$ This fuel is ejected straight downward with a velocity $V_{0}$
(a) Find $\beta$ as a function of time in order that the rocket can remain stationary in space a short distance off the ground.
(b) Assuming that the amount of fuel used per second remains constant at a value $\alpha$ but is greater than in $\langle a\rangle$, find the velocity of the rocket upward as a function of the time. Ans. With $M=M_{0}-\alpha t$ and $\alpha=d M / d t$, then $v=-g t+V_{0} \ln \left[M_{0} /\left(M_{0}-\alpha t\right)\right]$
(c) Compare this velocity for the case of $M=\frac{3}{4} M_{0}$ with that given in Eq. $(6.22) .$ Calculate these two velocities if $V_{0}=1.65 \times 10^{5} \mathrm{~cm} / \mathrm{s}$ and $\alpha=2 M_{0} g V_{0}$

Surendra Kumar

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Problem 15

Ice skaters retolving on end of a rope. Two ice skaters, each weighing $70 \mathrm{~kg}$, are traveling in opposite directions with speed $650 \mathrm{~cm} / \mathrm{s}$ but separated by a distance of $1000 \mathrm{~cm}$ perpendicular to their velocities. When they are just opposite each other, each grabs one end of a $1000-\mathrm{cm}$ -long rope.
(a) What is their angular momentum about the center of the rope before they grab the ends? After?
(b) Each now pulls in on his end of the rope until the length of the rope is $500 \mathrm{~cm}$. What is the speed of each?
(c) If the rope breaks just as they get to $500 \mathrm{~cm}$ apart, what mass would it hold up against the force of gravity?
(d) Calculate the work done by each skater in decreasing their separation and show that this is equal to his change in kinetic energy.

Surendra Kumar

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Problem 16

Slowing down of space cehicle. A space vehicle has mass $200 \mathrm{~kg}$ and eross-sectional area $2 \times 10^{4} \mathrm{~cm}^{2} .$ It travels in a region without appreciable gravitational field through a rarefied atmosphere whose mass density is $2 \times 10^{-15} \mathrm{~g} / \mathrm{ce}$ with initial speed $7.6 \times 10^{5} \mathrm{em} / \mathrm{s}$. (These would be approximate values for a satellite $500 \mathrm{~km}$ above the surface of the earth.) Assume that the conditions of the example on page 183 apply, that all the gas the space vehicle encounters sticks to it.
(a) Work out the value of $c$. Consider the mass of gas picked up in $1 \mathrm{~s}$. Ans. $c=4 \times 10^{-11} \mathrm{~g} / \mathrm{cm}$.
(b) Using the conservation of momentum $M v=M_{0} v_{0}$, find the differential equation for $v$ in terms of $t$ and constants. $\mathrm{Ans}, d v / d t=-c v^{3} / M_{0} v_{0^{-}}$
(c) Solve for $v$ and find the time required for the satellite to slow down to $0.9$ of its initial speed.

Surendra Kumar

Numerade Educator