Special Relativity: Dynamics

Problem : In the lab frame a particle has velocity (v _x, v _y) = (0.6c, 0) acceleration (a _x, a _y) = (2, 3) . What force is observed to be acting on the particle in the instantaneous inertial frame of the particle?

Solution for Problem 1 >>

First let us calculate the force in the lab frame. This is given by: (F _x, F _y) = m(γ ³ a _x, γa _y) , where γ = = 5/4 . Thus (F _y, F _x) = ((5/4)³×2,(5/4)*3) = (125/32, 15/4) . For a frame such as the particle's, moving along with speed 0.6c with respect to the lab we have F _x' = F _x and F _y' = γF _y . Thus the force in the instantaneous inertial rest frame of the particle is (F _x', F _y') = (125/32, 75/16) = 25/16(5/2, 3) .

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Problem : If p = γ ² mv _x , what would F _x be?

Solution for Problem 2 >>

We need the quantity F = = . First let us calculate :

= = = 2γ 4 va

Then:

F = = m( v + γ 2 ) = m(2γ 4 v 2 a + γ 2 a) = maγ(2v 2 γ 2 -1) = γ 3 ma(1 + v 2/c 2)

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Problem : Use the definition of force found in Section 3 and what you know about the relationship between force and energy to find an expression for the relativistic energy.

Solution for Problem 3 >>

We have Fdx = dE so integrating both sides:

Fdx = (γ 3 ma)dx
= γ 3 mv dx
= γmc 2|v2 v1

Which is ΔE for a particle changing from v ₁ to v ₂ . Of course, the energy change is offset by an equivalent change in potential energy such that total energy is conserved.

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Problem : Define a quantity b _i = . Let (b _x, b _y) = γ ³ , γ similar to the relation for force. How does this quantity transform between a stationary frame and a frame moving with speed v in the x -direction?

Solution for Problem 4 >>

We know that (b _x, b _y) = (γ ³ , γ ) . In the moving frame γ = 1 so: (b _x', b _y') = , . But because of time dilation and length contraction this becomes:

(b x', b y') = γ 3 , γ 3 = γ 4 , γ 3

Thus, b _x = and b _y = .

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Problem : In some frame F' which is moving at speed v in the x -direction with respect to frame F , there is a mass attached the end of a spring. The spring is constraining to vibrate along a straight line which lies at an angle θ' to the x' -direction in frame F' . In frame F , what is the direction of the spring oscillation, the direction of the acceleration of the mass, and the direction of force on the mass?

Solution for Problem 5 >>

In the stationary frame the frame F appears to be moving with speed v in the opposite direction. Hence, the projection of the spring oscillation on the x -axis must be reduced by a factor γ due to length contraction. Thus tanθ = γtanθ' . Since the mass is being pulled in the direction of the spring oscillation (this fact must be the same in any inertial frame), the acceleration must be in this same direction ( tanθ = γtanθ' ). From what we learned about force transformations between frames, the x -component of the force remains the same, but the y component is decreased by a factor γ ; thus, the angle between the horizontal and the force is μ , where tanμ = .

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