Assignment 1, solutions

Problem 1:

imageTwo masses are arranged as shown.  Mass m1 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 m2 = 6.00 kg, is sitting on the horizontal surface and is being pulled to the right by a force F, so that m2 is moving to the right at a constant speed.  The coefficient of sliding friction between m1 and m2 is μ1 = 0.380, while the coefficient of sliding friction between m2 and the horizontal surface is μ2 = 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 m2 to the right at a constant speed?


Problem 2:

(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.



Problem 3:

A particle of mass m is given an initial velocity v0i.  Assume that the particle is subject to a drag force F = -bv½i.
(a)  Find v as a function of time.
(b)  How far does the particle travel before coming to rest?


Problem 4:

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/s2 for the acceleration due to gravity.


Problem 5:

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 = 45o give a numerical answer.

MEarth = 5.97*1024 kg, REarth = 6378 km.