35. Special Relativity

According to the theory of relativity, how is time dilation (that is, $t=\frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$) perceived by an observer watching a moving clock compared to their own clock?

A 0.30-kg meter stick moving parallel to its length passes you at high speed. You measure its length to be 48.0 cm. What is its kinetic energy?

What are the rest energy, the kinetic energy, and the total energy of a 1.0 g particle with a speed of 0.80c?

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 1.5 years has elapsed? [Hint: Use the binomial expansion, Appendix A–2.]

Calculate the escape velocity, using Newtonian mechanics, from an object that has collapsed to its Schwarzschild radius.

Astronomers have measured the rotation of gas around a possible supermassive black hole of about 2 billion solar masses at the center of a galaxy. If the radius from the galactic center to the gas clouds is 68 light-years, estimate the value of z.

The Sun radiates energy at a rate of about 4 x 10²⁶ W.

(a) At what rate is the Sun’s mass decreasing?

(b) How long does it take for the Sun to lose a mass equal to that of Earth?

(c) Estimate how long the Sun could last if it radiated constantly at this rate.

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers on Earth?

In one of Thomson’s experiments he placed a thin metal foil in the electron beam and measured its temperature rise. Consider a cathode-ray tube in which electrons are accelerated through a 2000 V potential difference, then strike a 10 mg copper foil. What is the electron-beam current if the foil temperature rises 6.0°C in 10 s? Assume no loss of energy by radiation or other means. The specific heat of copper is 385 J/kg K .

Your job is to synchronize the clocks in a reference frame. You are going to do so by flashing a light at the origin at t = 0 s. To what time should the clock at (x, y, z) = (30 m, 40 m, 0 m) be preset?

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: What is the distance traveled according to observers on the spacecraft?

A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height?

The factor γ appears in many relativistic expressions. A value γ = 1.01 implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is γ = 1.01 for (a) an electron, (b) a proton, and (c) an alpha particle?

Use special relativity and Newton’s law of gravitation to show that a photon of mass m = E/c² just grazing the Sun will be deflected by an angle ∆θ given by ∆θ = 2GM/c²R, where G is the gravitational constant, R and M are the radius and mass of the Sun, and c is the speed of light. Put in values and show ∆θ = 0.87". (General Relativity predicts an angle twice as large, 1.74".)