Physics 341 Test 1

1.

λ₂=

1	0	0
0	0	1
0	-1	0

λ₆=

-1	0	0
0	-1	0
0	0	-1

Calculate det(λ₂) - det(λ₆).

2.

Show that

ε_ijkδ_ij = 0
ε_ijkε_ljk = 2δ_il
ε_ijkε_ijk = 6

3.

Find the angle between the surfaces defined by r²=9 and x + y + z² at the point (-2, 2, -1).

4.

(a) Using the knowledge that the total momentum of a rocket (mass m and velocity v) and its exhaust (mass dm' and velocity u) are the same before and after the rocket engine is fired, derive the rocket equation.

(b)Using the velocity from the rocket equation: v = v₀ + uln(m₀/m), find the mass ratio m/m₀ at the time when the rocket momentum (mv) reaches a maximum.

(c)Low Earth orbit (LEO) can be achieved when the centripetal force, mv²/r, balances the force of gravity, mg. Of course, g depends on height, but for LEO we aren't making too great an error to use the surface value. If the Earth's radius is taken as 6380km, and the shuttle main engine is capable of exhaust velocities of 2900m/s, what is the mass ratio of the payload to rocket if accelerating from rest to orbital speed? (Note, we are not rising vertically, nor are we taking into account air resistance.)

5.

A simple harmonic oscillator consists of a 100-g mass attached to a spring whose constant is 10⁴ dynes/cm. (A dyne is the unit of force in the cgs system=10^-5newtons) The mass is displaced 3 cm and released from rest. Calculate: (a) the natural frequency, ν, and the period, τ, (b) the total energy, and (c) the maximum speed. Please show your work.