Two masses are arranged
as shown. Mass m_{1} has a mass
of 4.00 kg and is attached to the vertical surface on the left with a string in
which the tension is T. Mass m_{2}
= 6.00 kg, is sitting on the horizontal surface and is being pulled to the right
by a force F, so that m_{2} is moving to the right at a constant speed.
The coefficient of sliding friction between m_{1} and m_{2}
is μ_{1} = 0.380, while the coefficient of sliding friction between m_{2}
and the horizontal surface is μ_{2} = 0.510.

(a)
What will be the tension T in the string?

(b)
What will be the magnitude of the force F required to pull m_{2}
to the right at a constant speed?Solution:

(a) Imagine
that the earth were of uniform density and that a tunnel was drilled along a
diameter. If an object were dropped into the tunnel, show that it would
oscillate with a period equal to the period of a satellite orbiting the earth
just at the surface.

(b) Find the
gravitational acceleration at a point P,
a distance x from the surface of
a spherical object of radius R.
The object has density ρ.
Inside the object is a spherical cavity of radius R/4. The center of this cavity is
situated a distance R/4 beyond
the center of the large sphere C,
on the line from P to C.

A particle of mass m is given an initial velocity v_{0}**i**.
Assume that the particle is subject to a drag force **F** = -bv^{½}**i**.

(a) Find v as a function of time.

(b) How far does the particle travel before coming to rest?

A thrill-seeking student jumps off a bridge from the height H = 80.0 m above the
water level. The student is attached to the bridge by an elastic cord so that
she reaches zero velocity just as she touches the water. After some up-and-down
bouncing, the student eventually comes to rest
h = 20.0 m above the water level. What is the maximum speed reached by the
student during the fall? Neglect air resistance and use 10.0 m/s^{2}
for the acceleration due to gravity.

Assume a perfectly spherical Earth of radius R with a
frictionless surface. On the surface of this Earth an object with mass m is
moving with constant speed v towards the north pole. When the object is at
latitude λ, find the external force required to keep it moving on that
trajectory.

For m = 1 kg, v = 500 m/s, λ = 45^{o} give a numerical answer.

M_{Earth} = 5.97*10^{24} kg, R_{Earth} = 6378 km.