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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 = √2d, 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 A1 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 A1.

(1)  As a result of traveling through Region A1, 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 A1, 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 A1 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 A1. What is the value of f? Assume that d is negligible compared to f. While P is traveling through Region A1, 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 I1 and for I2? 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 A1 described in Question I(2), there is another region, Region A2, 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 A1 and A2 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 A1 at y = y0, its trajectory is changed by a very small angle θ0 before entering Region A2. 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 A1 separately from Particle P at y = y0, its trajectory is changed by θ0/2 before entering Region A2.

(1)  What is the mass of Particle Q?

(2)  In multiples of y0, what are the y-coordinates at which Particle P and Particle Q enter Region A2?

(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 A2?  Write your answer in terms of k and θ0.

(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 f0. The thickness of Plate A is hA, and the velocity of a longitudinal ultrasonic wave propagating inside Plate A is VA. The plate surfaces are mechanically free.

(1)  Express f0 in terms of hA and VA.

(2)  When VA = 5.0 x 103 m/s, resonance occurred at frequencies of 2.0 x 106 Hz and 3.0 x 106 Hz. What is the minimum value of hA? 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. VB is the velocity of the longitudinal wave propagating inside Plate B, and VB/k > VA. Assume that 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 VA. 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　)　=VAT, PS = SP (shared). Therefore, ΔPQS and ΔSRP are congruent. Additionally, ΔPQS is a right triangle with PQS being the right angle, so 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, hA, VA, VB, α­, and ϕ’. Ignore the size of X. (End of the problems)