Chapter 4, Frames of Reference: Galilean Transformation Video Solutions, Mechanics Berkeley Physics

Problem 1

Block on rotating table. A block is to remain at rest relative to a rough horizontal table that is rotating at $20 \mathrm{rpm}$. The block is $150 \mathrm{~cm}$ from the axis of rotation that is vertical. How big must the coefficient of friction be? Show the centrifugal and the frictional forces on a diagram.

Khoobchandra Agrawal

Numerade Educator

07:17

Problem 2

Moving frame. In a railroad car traveling at $500 \mathrm{~cm} / \mathrm{s}$ along a straight track, a head-on collision takes place between a $100-g$ mass moving with velocity $100 \mathrm{~cm} / \mathrm{s}$ in the same direction as the train and a $50-\mathrm{g}$ mass moving in the opposite direction at $500 \mathrm{~cm} / \mathrm{s}$. Both velocities are relative to the train.After the collision, in the car the $50-g$ mass is at rest; what is the velocity of the $100-g$ mass? How much kinetic energy has been lost? $\quad$ Ans. $-150 \mathrm{~cm} / \mathrm{s}$. Now describe the collision from the point of view of an observer at rest by the track. Is momentum conserved? How much kinetic energy is lost in this frame?

Kevin Shryock

Numerade Educator

03:28

Problem 3

Acceleration in circular motion. An object moves in a circular path with a constant speed $v$ of $50 \mathrm{~cm} / \mathrm{s}$. The velocity vector $\mathbf{v}$ changes direction by $30^{\circ}$ in $2 \mathrm{~s}$.
(a) Find the magnitude of the change in velocity $\Delta \mathbf{v}$.
(b) Find the magnitude of the average acceleration during the interval. $\quad$
(c) What is the centripetal acceleration of the uniform circular motion? $\quad$

Khoobchandra Agrawal

Numerade Educator

00:49

Problem 4

Effective force due to rotation. An object fixed with respect to the surface of a planet identical in mass and radius to the earth experiences zero gravitational acceleration at the equator. What is the length of a day on that planet?

Khoobchandra Agrawal

Numerade Educator

01:18

Problem 5

Motion in a noninertial reference frame. Consider an inertial frame $S$ on the surface of the earth and a noninertial frame $S^{\prime}$ at rest in a freely falling elevator.
(a) What is the equation of motion in $S^{\prime}$ of a freely falling particle in $S ?$
(b) What are the applied and fictitious forces in $S$ and $S^{\prime}$ on the particle in $(a) ?$
(c) What are the equations of motion in $S^{\prime}$ of a particle moving in a horizontal circle in $S$ ? Assume $y=y^{\prime}=0$

Hunza Gilgit

Numerade Educator

00:55

Problem 6

Pendulum in accelerated car. A pendulum hangs vertically in a car at rest. At what angle will it hang when the acceleration of the car moving on a horizontal plane is $100 \mathrm{~cm} / \mathrm{s}^{2}$ ?

Khoobchandra Agrawal

Numerade Educator

01:52

Problem 7

Centrifuge for humans. In aviation medicine studies, centrifuges, which are horizontal shafts rotating about a vertical axis and carrying an experimental subject at one end of the shaft, are used. If the distance of the subject from the center of rotation is $700 \mathrm{~cm}$, how fast must the centrifuge rotate in order to subject the rider to $5 \mathrm{~g}$ ? $g=$ accel. of gravity.

Keshav Singh

Numerade Educator

13:12

Problem 8

Accelerated frame. A frame of reference has an upward acceleration of $300 \mathrm{~cm} / \mathrm{s}^{2}$. At $t=0$, its origin is at rest and $\mathrm{co}$ incident with that of an inertial frame on the surface of the earth. (Neglect the rotation of the earth.)
(a) Assuming $y$ is up and $x$ is horizontal, find $x(t)$ and $y(t)$ in both frames for an object that is projected horizontally with a speed of $1000 \mathrm{~cm} / \mathrm{s}$ at $t=0$, neglecting gravity.
(b) Work $(a)$ including gravity.

Sinisa Stura

Numerade Educator

02:45

Problem 9

Collision kinematics; center of mass. Two particles of mass $M_{1}=100 \mathrm{~g}$ and $M_{2}=40 \mathrm{~g}$ have initial velocities $\mathrm{v}_{1}=2.8 \hat{\mathrm{x}}-$
$3.0 \hat{\mathrm{y}} \mathrm{cm} / \mathrm{s}$ and $\mathbf{v}_{2}=7.5 \hat{\mathrm{y}} \mathrm{cm} / \mathrm{s}$. They collide, and after the
collision the velocities are $\mathbf{v}_{1}^{\prime}=1.2 \hat{\mathbf{x}}-2.0 \hat{\mathbf{y}} \mathrm{cm} / \mathrm{s} \quad$ and
$\mathbf{v}_{2}^{\prime}=4.0 \hat{\mathbf{x}}+5.0 \hat{\mathrm{y}} \mathrm{cm} / \mathrm{s}$
(a) Find the total momentum.
(b) Find the velocity of a reference frame in which the total momentum (before collision) is zero. This is called the center-of-mass frame.
(c) Show that the momentum is zero in this frame after the collision.
(d) What fraction of the initial kinetic energy is not present as kinetic energy after the collision? Is the collision elastic?

Nick Johnson

Numerade Educator

03:32

Problem 10

Unequal-mass collision. In the collision of two particles, the reference frame in which one is initially at rest and the other moving with velocity $v$ is called the laboratory frame. Suppose the moving mass is $m$ and the stationary one is $2 m$.
(a) What is the velocity of the center-of-mass frame (see Prob.
9) with respect to the laboratory frame?
(b) How much kinetic energy is lost in both the laboratory and center-of-mass frames in a completely inelastic collision, i.e., one in which the particles stick together?
(c) If the collision is elastic, the velocities of the particles in the center-of-mass frame are changed in direction but not in magnitude. Find an expression relating the angle of deviation (usually called angle of scattering) of the mass $m$ in the laboratory and in the center-of-mass frames.

Note that in the center-of-mass frame, the angle of the second particle is always $180^{\circ}$ from the angle of the first. In the equal-mass collision, $\theta_{\mathrm{lab}}=\theta_{\mathrm{c} . \mathrm{m}} / 2 .$ Vector diagrams are instructive.

James Kiss

Numerade Educator

02:01

Problem 11

Acceleration and magnetic deflection of electrons. (This problem and Probs. 12 to 14 are reviews of material in Chap.
3.) Suppose electrons are liberated at rest at point 0 on a metallic plane (see Fig. $4.21$ ) and are accelerated toward a parallel plane $0.25 \mathrm{~cm}$ away by an electric field. A tiny hole at $P$ permits a beam of electrons to escape into a region free of electric fields (all in a high vacuum, of course). The electric field is produced by applying voltages of $-300$ and $0 \mathrm{~V}$ to the metallic planes, as indicated. It is desired to bend the beam through a $90^{\circ}$ angle in a circular path of radius $0.5 \mathrm{~cm}$ by a magnetic field $B$ of circular outline as shown. Calculate the field strength required; also state its direction. (Note: An electric field intensity of 1 statvolt $/ \mathrm{cm}$ is equal to $300 \mathrm{~V} / \mathrm{cm}$.)

Dominador Tan

Numerade Educator

01:42

Problem 12

Transit time of ions. A pulse of singly charged cesium ions $\mathrm{Cs}^{+}$ is accelerated from rest by an electric field of 1 statvolt $/ \mathrm{cm}$ acting for $0.33 \mathrm{~cm}$ and afterward travels $1 \mathrm{~mm}$ in $87 \times 10^{-9} \mathrm{~s}$ in an evacuated field-free space.
(a) Derive from these data a value of the atomic mass of Cs $^{+} .$ Ans. $2.4 \times 10^{-22} \mathrm{~g}$.
Compare with the value you will find in tables, handbooks, or chemistry textbooks.
(b) What would be the time for protons to transit the $1-\mathrm{mm}$ region?

Chai Santi

Numerade Educator

01:49

Problem 13

13. Magnetic deflection of electron beam. Deflection of an electron beam in a cathode-ray tube may be accomplished by magnetic as well as by electrostatic means. A beam of electrons of energy $W$ enters a region of transverse uniform magnetic field of strength $B$. (Neglect fringe effects. See Fig. 4.22.)
(a) If $x$ is the distance from the point at which the electron entered the field to the point at which it leaves it, show that
$$
y=r\left[1-\sqrt{1-\left(\frac{x}{r}\right)^{2}}\right]
$$
where $r$ is the radius of curvature of the electron in the transverse magnetic field. The radius of curvature is the radius of the circle that will match (coincide with) the curved portion of the path.
(b) If $R$ is the radius of the magnet poles, then $x \approx 2 R$ when $r \gg R$. Use the binomial expansion to show $y \approx 2 R^{2} / r$.

Anand Jangid

Numerade Educator

02:34

Problem 14

Acceleration in a cyclotron. Suppose in a cyclotron that $\mathbf{B}=\hat{\mathbf{z}} B$ and
$$
E_{x}=E \cos \omega_{c} t \quad E_{y}=-E \sin \omega_{c} t \quad E_{z}=0
$$
with $E$ constant. (In an actual cyclotron the electric field is not uniform in space.) We see that the electric field intensity vector sweeps around a circle with angular frequency $\omega_{c^{*}}$. Show that the displacement of a particle is described by
$$
\begin{aligned}
x(t) &=\frac{q E}{M \omega_{c}^{2}}\left(\omega_{c} t \sin \omega_{c} t+\cos \omega_{c} t-1\right) \\
y(t) &=\frac{q E}{M \omega_{c}^{2}}\left(\omega_{c} t \cos \omega_{c} t-\sin \omega_{c} t\right)
\end{aligned}
$$
where at $t=0$ the particle is at rest at the origin. Sketch the first few cycles of the displacement.

Nick Johnson

Numerade Educator