4. Frame {2} is rotated with respect to frame {1} about x-axis by an angle of 60 degree. The position of the origin of frame {2} with respect to frame {1} is [6.0 5.0 7.0]". Find the homogeneous transformation matrices of frame {1} with respect
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![4. Frame {2} is rotated with respect to frame {1} about x-axis by
an angle of 60 degree. The position of the origin of frame {2}
with respect to frame {1} is [6.0 5.0 7.0]'
homogeneous transformation matrices of frame {1} with respect
to frame {2}.
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- Consider a point charge q at rest, situated at the origin of some frame S. Its electric field, according to S is 9 E(,1) = 1 2/21 47€ 12 Of course, its magnetic field is zero. Now, you want to know what will happen to the fields if q moved to the right (positive x direction). Since it is at rest in the S frame, what you can do is go to another frame, S', then boost it to the negative x direction. This way, q moves to the right according to S'. Let the velocity of S'be v = -vi Solve for the magnetic field components as seen in the S' frame. Combine these to write down B'(x, t)A spaceship leaves the solar system at v = (3/5)c and is headed towards a planet that is 20 c • years away (c is the speed of light). Assume the following: the Sun and the planet are mutually at rest and their clocks have been synchronized such that both read zero when the spaceship leaves. Say that the clock on the ship began at zero. If this is the case, then what should the clock on the ship read when it arrives at the planet?Please explain in detail An observer P stands on a train station platform as a high-speed train passes by at u/c = 0.8. The observer P, who measures the platform to be 60 m long, notices that the front and back ends of the train line up exactly with the ends of the platform at the same time. (a) How long does it take the train to pass P as he stands on the platform, as measured by his watch? (b) According to a rider T on the train, how long is the train? (c) According to a rider T on the train, what is the length of the train station platform?
- Suppose two angry rhinoceros charge each other. According to a few observers on the ground, the rhinoceros on the left moved to the right at a speed of (1 - ϵ)c while the rhinoceros on the right moved to the left at a speed of (1 - ϵ)c. ϵ is in the range of 0 < ϵ < 1. How fast is the rhinoceros on the right moving in the frame of the rhinoceros on the left? (Show that this speed is less than c, no matter how small ϵ is)Consider a particle that, once it comes to existence, decays on average after a short time t, where t is a time measured in the particle's frame of reference. If such a particle travels with speed v in an observer’s frame, what is the distance the particle has travelled as observed by the observer?Einstein concluded that the speed of light is the same in all inertial frames. Prove that the analysis of the Michelson-Morley experiment (see attached image) is in error.
- Calculate the interval ∆s2 between two events with coordinates ( x1 = 50 m, y1 = 0, z1 = 0,t1 = 1 µs) and (x2 = 120 m, y2 = 0, z2 = 0, t2 = 1.2 µs) in an inertial frame S. b) Now transform the coordinates of the events into the S'frame, which is travelling at 0.6calong the x-axis in a positive direction with respect to the frame S. Hence verify that thespacetime interval is invariant.Problem 2 In terms of the xs, ŷ, 2s coordinates of a fixed space frame {s}, frame {a} has its x-axis pointing in the direction (0, 0, 1) and its ŷ₂-axis pointing in the direction (-1,0, 0), and frame {b} has its x-axis pointing in the direction (1, 0, 0) and its y-axis pointing in the direction (0, 0, -1). The origin of {a} is at (3, 0, 0) in {s} and the origin of {b} is at (0, 2, 0) in {s}. (a) Draw by hand a diagram showing {a} and {b} relative to {s}. (b) Write down the rotation matrices Rsa and Rsb and the transformation matrices Tsa and Tsb. (c Calculate the matrix exponential corresponding to the exponential coordi- nates of rigid-body motion S0 = (0, 1, 2, 3, 0, 0). Draw the corresponding frame relative to {s}, as well as the screw axis S.Events 1 and 2 are exploding firecrackers that each emit light pulses. In the reference frame of the detector, event 1 leaves a char mark at a distance 3.40 m from the detector, and event 2 leaves a similar mark at a distance 2.10 m from the detector. If the two events are simultaneous in the reference frame of the detector and occur at instant t = 0, at what instant of time will each light pulse be detected?