Gravity

1.

Consider two masses suspended as shown in our Vancouver lab. Calculate the angle $\theta$ due to gravitational attraction. (Remember that $\theta$ will be very small.)

2.

Sitting in outer space, you arrange to place two identical 0.8 m diameter spheres of solid iron with their centres 3.2 m apart, as shown, initially at rest.

If only gravitational forces are acting, what will be the distance between their closest points 1 hr after you let them go?

3.

Someone in another planetary system has built a thin Ringworld with a diameter 3 times that of the Earth and with a mass equal to the Earth's. You approach it along its axis and decide to make your 1500 metric ton spaceship ``hover'' stationary at one Earth-radius from the ring's centre. What thrust (in Newtons) must your engines provide? What mass could be made to ``hover'' at the Earth's surface using this thrust?

4.

A 300 kg mass is placed at rest wrt the Earth at a location 4 Earth radii above the Earth's surface. By following the subsequent free-fall radially inwards [and ignoring air resistance in the atmosphere] with our step-by-step numerical approach, find the time to impact in minutes.

Note: apply the method judiciously; you are the arbiter of what degree of precision is reasonable to strive for within the constraints of your available computing power and affordable waking hours! Start with a crude estimate and then proceed.

If you know how to solve this exactly or wish to use energy considerations in some manner, that's fine; but you are requested to first solve the problem by our recommended method.