Special Relativity - Relativistic Energy

What is the speed of a particle having a momentum of 5 MeV/c and a total relativistic energy of 10 MeV?

A. c
B. 0.75 c
C. 1/√3 c
D. ½ c
E.  ¼ c

(GR9677 #38)

Solution:

p = γm₀v
E = γm₀c²

p/E = v/c²
v = pc²/E = (5 MeV/c) c²/10 MeV = ½ c

Answer: D

Special Relativity - Time Dilation

Tau leptons are observed to have an average half-life of Δt₁ in the frame S₁ in which the leptons are at rest. In an inertial frame S₂, which is moving at a speed v₁₂ relative to S₁, the leptons are observed to have an average half-life of Δt₂. In another inertial reference frame S₃, which is moving at a speed v₁₃ relative to S₁ and v₂₃ relative to S₂, the leptons have an observed half-life of Δt₃. Which of the following is a correct relationship among two of the half lives, Δt₁, Δt₂, and Δt₃.

A.

B.

C.

D.

E.

(GR9277 #38)

Solution:

Time Dilation:

Δt_j =  γ_ijΔt_i



Δt_i = time in a rest frame
Δt_j = time in a moving frame

In this problem, Δt₁ = time in a rest frame
Time dilation only relates to time in a rest frame Δt₁ with time in a moving frame, Δt₂ or Δt₃

The only right answers:
Δt₂ =  γ₁₂Δt₁
Δt₃ =  γ₁₃Δt₁

(C) and (D) are FALSE (no Δt₁ in the equation)
(E) is FALSE (relates v₂₃ with Δt₁ and Δt₂)

(A) FALSE
Δt₂ = Δt₁ [1 − (v₁₂)²/c²]^½ = Δt₁ (1/γ₁₂) → Δt₁ = γ₁₂Δt₂ 

(B) TRUE
Δt₁ = Δt₃ [1 − (v₁₃)²/c²]^½ = Δt₃ (1/γ₁₃) → Δt₃ = γ₁₃Δt₁ 

Answer: B

Electromagnetism - Logic Circuit

If logical 0 is 0 volts and logical 1 is +1 volt, the circuit shown above is a logic circuit commonly known as

A. an OR gate
B. an AND gate
C. a 2-bit adder
D. a flip-flop
E. a fanout

(GR8677 #38)

Solution:

Diode:




The above logic circuit represents an OR gate where the inputs are connected to the anodes of diodes and the cathodes to the output.

For an AND gate, the diodes are reserved. The cathodes are connected to the inputs and anodes to the output.

Answer: A

Electromagnetism - RLC Circuit

An AC circuit consists of the elements shown above, with R = 10,000 ohms, L = 25 millihenrie and C an adjustable capacitance. The AC voltage generator supplies a signal with amplitude of 40 volts and angular frequency of 1,000 radians per second. For what value of C is the amplitude of the current maximized?

A. 4 nF
B. 40 nF
C. 4 μF
D. 40 μF
E. 400 μF

(GR0177 #38)

Solution:

I_max when X_L = X_C

→ ωL = 1 / ωC

C = 1 / ω²L 
= 1 / [(10³)² × 25 × 10⁻³]
= 1 / (25 × 10³)
= 40 × 10⁻⁶ 
= 40 μF

Answer: D