【GR0877】 The Solution for GRE Physics Practice Test [81-100]

The Solution for GRE Physics Practice Test [81-100]

 

추천글 : 【GRE】 Solution for GRE Physics Practice Test 

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81. A uniform disk with a mass of m and a radius of r rolls without slipping along a horizontal surface and ramp, as shown below. The disk has an initial velocity of v. What is the maximum height h to which the center of mass of the disk rises?

 

 

⑴ answer: B

⑵ E = 1/2 mv² + 1/2 Iω² = 3/4 mv² = mgh 

 

 

82. A mass, m, is attached to a massless spring fixed at one end. The mass is confined to move in a horizontal plane, and its position is given by the polar coordinates r and θ. Both r and θ can vary. If the relaxed length of the spring is s and the force constant is k, what is Lagrangian, L, for the system?

 

 

⑴ answer: D

⑵ L (Lagrangian) = T (kinetic energy) - U (potential energy)

 

 

83. A mass m attached to the end of a massless rod of length L is free to swing below the plane of support, as shown in the figure below. The Hamiltonian for this system is given by 

 

 

where θ and ϕ are defined as shown in the figure. On the basis of Hamilton's equations of motion, the generalized coordinate or momentum that is a constant in time is 

 

 

⑴ answer: E

⑵ H = Σ p_i q_i' - L

① p_i : generalized momenta 

② q_i : generalized coordinate

⑶ External gravitation cannot affect the rate of ϕ.

 

 

84. A rod of length L and mass M is placed along the x-axis with one end at the origin, as shown in the figure below. The rod has linear mass density λ = 2Mx / L², where x is the distance from the origin. Which of the followig gives the x-coordinate of the rod's center of mass?

 

 

⑴ answer: E

⑵ x_CM = 1/M × ∫ λx dx = 1/M × ∫ 2Mx² / L² dx = 2L/3

 

 

85. A particle is in an infinite square well potential with walls at x = 0 and x = L. If the particle is in the state ψ(x) = A sin (3πx / L), where A is a constant, what is the probability that the particle is between x = L/3 and x = 2L/3?

 

 

⑴ answer: B

⑵ normalization : ∫ ψ² dx (from x = 0 to x = L) = 1 = LA² / 2 ⇔ A = √ (2 / L)

⑶ given probability : ∫ ψ² dx (from x = L/3 to x = 2L/3) = 1/3 

 

 

86. Which of the following are the eigenvalues of the Hermitian matrix 

 

 

⑴ answer: B

⑵ eigen value : det (H - λI) = 0

 

 

87. Consider the Pauli spin matrices σ_x, σ_y, and σ_z an the identity matrix I given below. The commutator [σ_x, σ_y] ≡ σ_xσ_y - σ_yσ_x is equal to which of the following?

 

 

⑴ answer: D

⑵ simple calculation 

 

 

88. A spin-1/2 particle is in a state described by the spinor 

 

 

where A is a normalization constant. The probability of finding the particle with spin projection S_z = -1/2 ℏ is

 

 

⑴ answer: D

 

 

89. An electron with total energy E in the region x < 0 is moving in the +x-direction. It encounters a step potential at x = 0. The wave function for x ≤ 0 is given by

 

 

and the wave function for x > 0 is given by

 

 

Which of the following gives the reflection coefficient for the system?

 

 

 

⑴ answer: D

⑵ The reflection coefficient of electromagnetic wave (η : intrinsic impedance)

 

 

⑶ The reflection coefficient of a material wave must be similar to the reflection coefficient of an electromagnetic wave by symmetry.

 

 

90. Two thin, concentric, spherical conducting shells are arranged as shown in the figure below. The inner shell has radius a, charge +Q, and is at zero electric potential. The outer shell has radius b and charge -Q. If r is the radial distance from the center of the spheres, what is the electric potential in ragion Ⅰ (a < r < b) and in region Ⅱ (r > b)?

 

 

⑴ answer: D

⑵ V_r - V_a = V_r = ∫ -kQ / x² dx (x = a to r) = kQ (1/r - 1/a) 

 

 

91. In static electromagnetism, let E, B, J, and ρ be the electric field, magnetic field, current density, and charge density, respectively. Which of the following conditions allows the electric field to be written in the form E = -∇ϕ, where ϕ is the electrostatic potential?

 

 

⑴ answer: C

⑵ Green theorem

 

 

92. A long, straight, hollow cylindrical wire with an inner radius R and an outer radius 2R carries a uniform current density. Which of the following graphs best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?

 

 

⑴ answer: E

⑵ R < r < 2R : B = μ₀ / 2πr × σ(πr² - πR²)

⑶ 3R < r : μ₀ / 2πr × 4πσ R²

 

 

93. A parallel-plate capacitor has plate separation d. The space between the plates is empty. A battery supplying voltage V₀ is connected across the capacitor, resulting in electromagnetic energy U₀ stored in the capacitor. A dielectric, of dielectric constant κ, is inserted so that it just fills the space between the plates. If the battery is still connected, what are the electric field E and the energy U stored in the dielectric, in terms of V₀ and U₀?

 

 

⑴ answer: B

⑵ C = κC₀ 

 

 

94. An observer O at rest midway between two sources of light at x = 0 and x = 10 m observe the two sources to flash simultaneously. According to a second observer O', moving at a constant speed parallel to the x-axis, one source of light flashes 13 ns before the other. Which of the following gives the speed of O' relative to O?

 

 

⑴ answer: C (🤔 there may be an error in it...)

⑵ Lorentz transformation for time : t' = γ(t - vx / c2)

⑶ Δt' = 13 × 10-9 = γ × v × 10 / c2 ⇔ v = 0.4325624c or 0.901604c 

 

 

95. Let Ĵ be a quantum mechanical angular momentum operator. The commutator [Ĵ_x, Ĵ_y, Ĵ_z] is equivalent to which of the following?

 

 

⑴ answer: D

 

 

 

96. Which of the following ions CANNOT be used as a dopant in germanium to make an n-type semiconductor?

 

 

⑴ answer: D

⑵ semiconductor : B can be used for an p-type semiconductor.

 

 

97. In the Compton effect, a photon with energy E scatters through a 90° angle from a stationary electron of mass m. The energy of the scattered photon is 

 

 

⑴ answer: E

⑵ Compton effect

 

 

⑶ λ' / ch - λ / ch = 1/E' - 1/E = 1/mc² ⇔ E' = E∙mc² / (E + mc²)

 

 

98. Which of the following is the principal decay mode of the positive muon μ⁺? 

 

 

⑴ answer: D

 

 

99. A small particle of mass m is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius r from the axis, as shown in the figure below. The platform begins to rotate with constant angular acceleration α. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed ω, the angle θ between the static frictional force f_s and the inward radial direction is given by which of the following?

 

 

⑴ answer: E

⑵ In the rotating frame of the particle, three forces are in equilibrium: centrifugal force, frinctional force, and inertial force due to angular acceleration.

 

 

100. The partition function Z in statistical mechanics can be written as 

 

 

where the index r ranges over all possible microstates of a system and E_r is the energy of microstate r. For a single quantum mechanical harmonic oscillator with energies

 

 

the partition function Z is given by which of the following? 

 

 

⑴ answer: E

⑵ Let's remember the shape of the denominator.

 

입력: 2023.10.07 21:10