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### The University of Tokyo 2013 Entrance Examination—Physics

**Question 1**

Answer the following questions **I** and **II**.

**I**** ** As
shown in Figure 1-1, two identical springs with a spring constant of *k* are placed so that they are directly
facing each other on a smooth, flat surface. For each of the springs, one end is
fixed to a rigid wall, while the other end is attached to a small ball with
mass *m* (Ball 1 for the left spring
and Ball 2 for the right spring). When the springs are at their natural length,
the distance between the two balls is *d*.
The size of the small balls and the mass of the springs can be ignored.

Now, as
shown in Figure 1-2, Ball 1 is moved from the equilibrium position so that the
left spring is compressed by a distance *s
*(*s* > *d*), and then released. Answer the following questions.

**(1) **Ball 1 starts to move and
collides with Ball 2. What is the velocity of Ball 1 right before the
collision?

**(2)** Assuming that the
collision between the two balls is an elastic collision, what are the
velocities of Ball 1 and Ball 2 right after the collision?

**(3)** After the collision, what
are the maximum compression distances from equilibrium for the spring with Ball
1 and the spring with Ball 2 before the balls collide again?

**(4)** If *s* = √2*d*, what is the time
between the first and second collisions?

**II**

**Next, consider the same springs and balls from Question I, except they are now on a rough, flat surface. The coefficients of static and kinetic friction between each of the balls and the surface are**

*µ*and

*µ’*, respectively. The gravitational acceleration is

*g*. Answer the following questions.

**(1)**** ** As in Question I (Figure
1-2), Ball 1 is moved a distance *s*
from its equilibrium position in the direction of compression, and then
released. What condition must be satisfied in order for Ball 1 to start moving
after being released?

**(2)** If the above condition is
satisfied, what is the minimum value *s*
can take for Ball 1 to collide with Ball 2?

**Question 2**

Consider controlling the motion of a charged particle using a
magnetic field. Assume that the effects of gravity can be ignored in answering
the questions below. The angles must be given in radians. For very small
angles, the following small-angle approximations can be used: cos *θ* ≒ 1, sin *θ* ≒ *θ*, tan *θ* ≒ *θ*.

**I**** ** As
shown in Figure 2-1, a magnetic field is directed along the *z*-axis, only in Region A_{1}
where |*x*|≤ *d*/2. (The *z*-axis is
pointing outwards from the plane of the paper towards you.) The magnetic flux
density varies slowly in the *y*-direction.
Particle P with mass *m* and positive
charge *q* is traveling in the positive
*x*-direction with velocity *v* as it enters Region A_{1}.

**(1)**** ** As a result
of traveling through Region A_{1}, Particle P’s direction of travel is
slightly altered to be a very small angle *θ*
from the *x*-axis. While P is traveling through Region A_{1},
the change in position in the *y*-direction
is minimal, so the magnetic flux density *B*
acting on P can be treated as being constant. What is the value of *θ*? From here on after, the values of
angles are positive in the direction of the arrow as indicated in Figure 2-1.

**(2)**** ** If the
magnetic flux density in Region A_{1} is proportional to the *y*-position value and can be expressed as
*B* = *by*, where *b* is a positive
constant, Particle P always travels through point (*x*, *y*) = (*f*, 0) on the *x*-axis, independent of the *y*-position
at which P enters Region A_{1}. What is the value of *f*? Assume that *d* is negligible compared to *f*.
While P is traveling through Region A_{1}, the change in position in
the *y*-direction is minimal, so the
magnetic flux density *B* acting on P
can be treated as being constant.

**(3)**** ** A set of
electromagnets is arranged as shown in Figure 2-2(a). An enlarged picture of
the area of concern, marked by a dotted square, is shown in Figure 2-2(b) along
with the coordinates. When the iron cores are shaped appropriately, a magnetic
field like the one presented in (2) can be realized in the plane where *z* = 0. In this case, what should the
direction of electric current be for I_{1} and for I_{2}? Write
down the sign for each. The arrows in the figure are pointing in the positive
direction.

**II**** ** Next, consider
the case where in addition to Region A_{1} described in Question I(2), there
is another region, Region A_{2}, centered around *x* = (3/2)*f* with width *d* in which a magnetic field is directed
along the *z*-axis (Figure 2-3). The
magnetic flux density of this magnetic field is *kby*, where *k* is a
constant. Assume that the change in angle of the direction of travel of a
particle that has traveled through both Regions A_{1} and A_{2}
can be calculated as the sum of the changes in angle, as determined in Question
I(1), after traveling through each of the regions. Also assume that *d* is negligible compared to *f*. When Particle P, traveling in the
positive *x*-direction with velocity *v*, enters Region A_{1} at *y* = *y _{0}*,
its trajectory is changed by a very small angle

*θ*before entering Region A

_{0}_{2}. When another particle, Particle Q, with the same electric charge

*q*as P and also traveling in the positive

*x*-direction with velocity

*v*, enters Region A

_{1}separately from Particle P at

*y*=

*y*, its trajectory is changed by

_{0}*θ*/2 before entering Region A

_{0}_{2}.

**(1)**** ** What is the
mass of Particle Q?

**(2)** In multiples
of *y _{0}*, what are the

*y*-coordinates at which Particle P and Particle Q enter Region A

_{2}?

**(3)**** ** What is the
angle between the *x*-axis and the
velocity vector for Particle P and Particle Q after they have traveled through
Region A_{2}? Write your answer in
terms of *k* and *θ _{0}*.

**(4)**** ** By tuning the
value of *k*, one can make Particle P
and Particle Q pass through the same point on the *x*-axis where *x* > (3/2)*f*. What is the value of *k* at which this will occur?

**Question 3**

Answer the following questions **I**, **II**, and
**III**. The angles must be given in radians.

**I**** ** As
shown in Figure 3-1, an ultrasonic oscillator is used to generate an ultrasonic
wave close to a plane wave in Plate A. The straight lines inside the plate
represent the wavefronts of the wave. When a longitudinal ultrasonic wave is
directed perpendicular to the surface of the plate with varying frequencies,
the plate resonates when the frequency is an integer multiple of *f _{0}*. The thickness of Plate A
is

*h*, and the velocity of a longitudinal ultrasonic wave propagating inside Plate A is

_{A}*V*. The plate surfaces are mechanically free.

_{A}

**(1)**** **
Express *f _{0}* in terms
of

*h*and

_{A}*V*.

_{A}**(2)** When *V _{A}* = 5.0 x 10

^{3}m/s, resonance occurred at frequencies of 2.0 x 10

^{6}Hz and 3.0 x 10

^{6}Hz. What is the minimum value of

*h*?

_{A}

**II** Both
longitudinal and transverse waves exist inside a solid. The velocities of the
two waves are different; the longitudinal wave is *k* times faster (*k* >
1). As shown in Figure 3-2, a two-layered plate is constructed by attaching
Plate A to Plate B, which is made from a different material. *V _{B}* is the velocity of the
longitudinal wave propagating inside Plate B, and

*V*>

_{B}/k*V*. Assume that

_{A}*k*does not depend on the material.

As
shown in Figure 3-2, a longitudinal wave is introduced into Plate A at an angle
*α* (0 < *α* < π/2) from point O on the surface of Plate A. As a result,
reflected and refracted longitudinal waves, as well as reflected and refracted
transverse waves, are generated at the boundary layer. The angles of reflection
for the longitudinal and transverse waves are *θ* and *θ’*, respectively. The
angles of refraction for the longitudinal and transverse waves are *ϕ* and *ϕ’*, respectively.

** (1)**** ** Based on Huygens’ principle, the angle of
reflection *θ* and the angle of
incidence *α* of the longitudinal wave
are equal. Using P, Q, R, and S as shown in Figure 3-3, fill in the blanks (a-d) in
the following passage.

In
Figure 3-3, an incident wave with a wavefront parallel to PQ is propagating
with velocity *V _{A}*. After
time T from the time at which two points on the wavefront pass P and Q, the
side that passed Q reaches point S on the boundary layer. At the same time, RS,
which is tangent to the semi-circle formed by the elementary wave generated
from P, becomes the wavefront of the reflected wave. Regarding ΔPQS and ΔSRP, ∠PQS =∠SRP =π/2,

**( a )**=

**( b )**=

*V*T, PS = SP (shared). Therefore, ΔPQS and ΔSRP are congruent. Additionally, ΔPQS is a right triangle with ∠PQS being the right angle, so

_{A}*α*= ∠

**( c )**. Similarly, ΔSRP is a right triangle with ∠SRP being the right angle, so

*θ*= ∠

**( d )**.Therefore,

*α*=

*θ*.

**(2)** Regarding the angle of
reflection of the transverse wave, *θ’*,
what is sin *θ’ *?

**(3)**** ** Regarding the
angle of refraction of the longitudinal wave, *ϕ*, and the angle of
refraction of the transverse wave, *ϕ’*, what are sin *ϕ* and sin *ϕ’* ?

**III** In
the two-layered plate constructed in II, a foreign matter X exists at a depth *h* from the boundary layer. As shown in
Figure 3-4, an ultrasonic wave is generated from O, and *t* is the time it takes for the wave reflected off the surface of X
to come back to O. Let us consider how to determine *h* from the measured value of *t*.

**(1)** First, we
would like to tune the angle of incidence *α*
so that the refracted wave propagating through Plate B is only a transverse
wave. What is the condition that sin *α*
must satisfy for this to occur?

**(2)** When a
longitudinal wave is introduced from O with an angle of incidence *α* that satisfies the condition
determined in (1), a transverse wave is generated within Plate B from point Y
in the boundary layer with an angle of refraction of *ϕ’*. After the wave
reaches X, a reflected wave travels back the same path to O. Express *t* in terms of *k*, *h*, *h _{A}*,

*V*,

_{A}*V*,

_{B}*α*, and

*ϕ’*. Ignore the size of X.

**(End of the problems)**

***The answers are available and please confirm if necessary.**