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### Nagoya University Entrance Examination (Year 2012) :Physics

*Physics exam time limit: approximately 75 minutes; *A total of 150 minutes is allocated for the combination of the Physics exam and the other science category exam.

#### Physics Problem I

As shown in Fig. 1, consider the motion of a small body fired by a spring. The mass of the small body is *m*, and the size can be neglected. One end of the spring is fixed, and the other end is attached to a plate. The spring follows Hooke’s law with a spring constant *k*. The air resistance, mass of the spring, and mass of the plate can be neglected. Gravity acts vertically downward with a magnitude equal to the gravitational acceleration, *g*. All motion occurs in the vertical plane, as shown in Fig. 1. Answer the following questions.

The small body is initially pushed so that the spring is compressed by a distance *d* relative to its equilibrium length. The small body is then gently released. The small body moves away from the plate toward the right along the floor with a speed *v*_{0}. The friction between the floor and the small body can be neglected.

Question (1): Express the speed *v*_{0} in terms of the appropriate parameters, chosen from *m, d, g,* and *k*.

A horizontal ditch is present on the floor. A cart with mass *M* is stationary and positioned at the left-hand side of the ditch. The top surface of the cart is flat and is collinear with the line of the floor (at the same height).

After the small body moves onto the cart from the floor, both the small body and the cart move to the right. The speed of the small body at the moment it moves onto the cart is *v*_{0}. Friction is present between the top surface of the cart and the small body with a dynamic friction coefficient *μ*′. Both the right edge of the cart and the small body simultaneously reach the right edge of the ditch. The friction between the cart and the ditch can be neglected.

Question (2): As the small body moves along the surface of the cart, the acceleration (assume the positive direction is to the right) of the small body and the cart are *a* and *A*, respectively. Express the respective equations of motion of the small body and the cart in terms of the appropriate parameters, chosen from *m, M, v*_{0}, *a, A, g, *and *μ*′.

Question (3): Express the speed *v*_{1} in terms of the appropriate parameters, chosen from *m, M, v*_{0}*, g,* and *μ*′.

The curved surface PQ is smoothly connected to the cylindrical surface QS. The center and radius of the cylindrical surface QS are O and r, respectively. The axis of the cylinder is perpendicular to the vertical plane in which the small body moves. The points P, O, and S are collinear with the line of the floor (at the same height). The highest point of the cylindrical surface QS is R. Here ∠QOS = *θ* (*θ* > 90°).

The small body moves from the cart to the curved surface and reaches the point R without detaching from the curved surface or the cylindrical surface. Upon passing the point Q, the speed of the small body on the cylindrical surface is *v*_{2}. N is the magnitude of the normal force applied to the small body by the cylindrical surface. *v*_{3} is the speed of the small body at the point R. The friction between the curved surface and the small body and the friction between the cylindrical surface and the small body can be neglected.

Question (4): Express the speed v3 in terms of the appropriate parameters, chosen from *m, M, r, θ, v _{2},* and

*g*.

Question (5): Express the magnitude of the normal force N in terms of the appropriate parameters, chosen from *m, M, r, θ, v _{2}, *and

*g*.

Question (6): Express the conditions on v2 under which the small body can reach the highest point R without detaching from the cylindrical surface in terms of the appropriate parameters, chosen from *m, M, r, θ,* and *g*.

#### Physics Problem II

As shown in Fig. 1, an undeformable rectangular coil JKLM (one turn) composed of a uniform conducting wire is placed in the *xz*-plane. The mass of the coil is *m*, and the coil experiences a gravitational force toward the +*z* direction (vertically downward). The magnitude of the gravitational acceleration is *g*. Each side of the coil is parallel to either the *x*-axis or the *z*-axis. Side *a* is parallel to the *x*-axis and side *b* is parallel to the *z*-axis. The *z*-coordinate of the center of the coil is a variable, Z, and Z > *b*/2.

A magnetic field is applied in the +*y* direction (perpendicular to the sheet and toward the reader), and the magnetic flux density may be expressed as a function of only the *z*-coordinate, B = *hz*, where *h* is a positive constant, as shown in Fig. 2.

The coil, which is initially fixed, begins to fall once a latch is released. The coil moves only in the *xz*-plane and only toward the +*z*-direction without rotating. The speed and acceleration of the coil in the +*z*-direction at any point in time are *v* and *A*, respectively. The air resistance can be neglected. An electron’s charge is -*e* and the electric resistance of the coil is *R*. The effects of the magnetic field produced by the electric current flowing through the coil can be neglected. Answer the following questions.

Question (1): Provide the appropriate terms or numerical expressions for (a) through (f).

Directions may be indicated as “+x-direction” or “-z-direction”. Numerical expressions may be expressed in terms of the appropriate parameters, chosen from *m, g, e, R, h, a, b, Z, v, *and *A*.

The magnetic flux density *B* on a side of the coil, JK, can be expressed as (a). An electron in one side of the coil, JK, experiences a force of magnitude(b) toward the (c)-direction due to the motion of the electron (velocity *v*) toward the +*z*-direction. This force is called “(d)”. An electron in one side of the coil, ML, experiences a (d) force with a magnitude (e) in the (f)-direction.

Question (2): Express the magnitude of the electromotive force, *V*, induced in the coil and the magnitude of the current, *I*, flowing in the coil in terms of the appropriate parameters, chosen from *m, g, e, R, h, a, b, Z, v,* and *A*. Also, answer the question; In which direction does the electric current flow: JKLM or MLKJ?

Question (3): As the coil falls with a speed *v*, show that the force *F* in the *z*-direction, acting on the coil due to the magnetic field, which causes current to flow through the coil, is expressed as:

Question (4): Express the equation of motion of the coil in the *z*-direction in terms of the appropriate parameters, chosen from *m, g, e, R, h, a, b, Z, v, *and *A*.

Question (5): After the coil has fallen for some period of time, the speed of the coil reaches a constant value. Under these conditions, express the electric current *I*_{1} flowing through the coil in terms of the appropriate parameters, chosen from *m, g, e, R, h, a, b, *and *Z*.

Question (6): Under the conditions that the coil falls at a constant speed, express the Joule heat *Q* generated in the coil per unit time and the power *W* applied to the coil under the force of gravity in terms of the appropriate parameters, chosen from *m, g, e, R, h, a, b, *and *Z*.

#### Physics Problem III

The thickness of a thin film on a glass plate may be measured experimentally using light and a diffraction grating, as explained in the following. The refractive indices of air, the thin film, and the glass are n_{o} = 1.0, n_{1} = 1.5, and n_{2} = 1.7, respectively. Visible light is light that has a wavelength between 4.0 × 10^{-7} m and 8.0 × 10^{-7} m. Answer the following questions.

Initially, as shown in Fig. 1, a parallel beam of light is vertically applied to the thin film, and interference is observed between the light reflected from the top surface and the light reflected from the bottom surface.

Question (1): The wavelength of the incident light in air is *λ*. Express the values of *λ* for which the two reflected beams of light (a) constructively interfere or (b) destructively interfere, in terms of the thickness *α* (> 0), the refractive index n1 of the thin film, and an integer m. Express the condition that m should satisfy in this case. Here, attention should be paid to the fact that as light enters a medium with a smaller refractive index from a medium with a larger refractive index, the phase of the reflected light reverses (the phase shifts by π) at the interface between the two media.

Question (2): The thickness of the thin film, *α*, is set to 5.0 × 10^{-7} m. If white light is directed onto the thin film, list all wavelengths *λ* [m] at which the two reflected beams constructively interfere, which would be observed as higher intensity reflected light at this wavelength.

Next, the diffraction grating is positioned such that light is applied vertically with respect to the diffraction grating surface, and transmitted light is observed on a screen positioned a distance *L* from the diffraction grating, as shown in Fig. 2. The screen is oriented perpendicular to the direction of propagation of the incident light. The slits in the diffraction grating are aligned perpendicular to the plane defined by the incident and refracted light. The angle between the incident light and the diffracted light (diffraction angle) is set to *θ*, the intersection of the incident light with the screen is O, and the distance between O and the point at which the diffracted light impinges on the screen is *x*.

Question (3): Provide the appropriate numerical expressions or values for (a) through (c).

The diffraction grating constant is set to *d*, where *L* >> *d*. The wavelength of light that enters the diffraction grating is *λ*. The conditions under which bright spots appear in the *θ*-direction can be expressed as (a) = *nλ* (where *n* = 0, ±1, ±2, ...) using *d* and *θ*.

Assuming that |*θ*| is sufficiently small such that the approximation sin*θ* ≒ tan*θ* holds, the distance between two adjacent bright spots on the screen, Δx, can be expressed as Δx =

(b) using *L*, *λ*, and *d*.

Setting *L* to 50 cm and directing monochromatic light with a wavelength of 5.0 × 10^{-7} m onto the diffraction grating yields bright spots at 5.0 cm intervals. Using Δx = (b), the number of slits present per 1 mm of the diffraction grating is (c).

In the last step, the thickness of a thin film may be obtained by positioning the diffraction grating in the light path of light reflected from the thin film, and the screen is positioned 50 cm from the diffraction grating, as shown in Fig. 3. The planes of the diffraction grating and screen are perpendicular to the path of the reflected light, and the slits of the diffraction grating are perpendicular to the plane defined by the incident and refracted light, as in the case shown in Fig. 2. White light is directed vertically onto the thin film, and the reflected light is observed on the screen after passing through the diffraction grating.

Question (4): In addition to the bright spot at point O, bright spots with varying brightness are observed at *x* = 4.5 cm and *x* = 6.0 cm in the vicinity of the point O. No other bright spots are observed between *x* = 6.0 cm and O. Based on this observation, calculate the thickness of the thin film, *a* [m], using the results of Question (3) by assuming that the diffraction angle is small. Assume that the observed bright spots are visible light.