UIUC Physics 211 James Scholars Problem Set 3

## Problem 1

Consider a block of mass M that is pulled up an incline by a force T that is parallel to the surface of the incline.   The block starts from rest and is pulled a distance x by the force T. The incline, which is frictionless, makes an angle theta with respect to the horizontal.

1.1  Write down the work done by the force T.

1.2  Calculate the potential energy of the block as a function of the position x. From the work and the potential energy calculate the kinetic energy of the block as a function of position x.

1.3  Calculate the acceleration, a, of the block as a function of position x. Calculate the velocity of the block from the acceleration and hence the kinetic energy as a function of position. Show that the kinetic energies calculated in these two ways are equivalent.

1.4  Calculate the ratio of the potential to kinetic energy. What happens to this ratio for large T? Check your answer for the case theta = pi/2 which you should be able to recalculate easily.

## Problem 2

A force in the xy plane is given by F_vec = (F_0/r)*(y*i^hat - x*j^hat), where F_0 is a constant and r is the distance from the origin: r2 = x2 + y2. This force acts on a particle of mass M constrained to move on a circle of radius r_0.

2.1  Show that the magnitude of the force is F_0 and that it is perpendicular to r_vec = x*i^hat + y*j^hat.

2.2  Calculate the work done by the force if the particle moves CW around its circular path for one orbit. What is the work done if it moves CCW around its path for one orbit?

2.3  Calculate the curl (look this up in a math book - it involves derivatives) of this force and show that it is non-zero. This non-zero curl is the signature of either a conservative or non-conservative force. Based on the answer to the previous question, what type of force is this?

2.4  Calculate the time, T_1, it takes the particle for its first orbit under the influence of this force. Calculate the time, T_2, it takes for its second orbit in terms of T_1.

## Problem 3

A block of mass M is dropped onto the top of a vertical spring whose force constant is k. The block is released from a height H above the top of the (relaxed) spring. Define u to be the displacement of the spring below its equilibrium point (i.e. when the block is in contact with it).

3.1  Find the value of the spring displacement, u, for the condition of maximum kinetic energy of the block.

3.2  In this question we will graph the kinetic energy vs. displacement. One of the ways used to draw general conclusions from specific situations is to use appropriate combinations of variables. First, write the expression for the kinetic energy, K, in terms of M, g, H, k and u. Next, rewrite the expression as a ratio: kappa = K/(MgH). Note that this ratio is dimensionless; when kappa = 1, K = MgH, etc. so it is quite easy to interpret. Another way of thinking about kappa is that it 'measures' K in units of MgH. Next substitute in your expression for kappa the (again dimensionless) variable w = u/H. Graph the 'reduced' kinetic energy, kappa, as a function of w. There will be one other dimensionless combination appearing in your result: kH/Mg which relates the spring force to the weight. Make your graph for kH/Mg = 3. Verify that the maximum value of kappa appears in the correct place.

3.3  What is the displacement corresponding to the maximum compression of the spring? Write your answer in terms of w = u/H as defined above. Give a numerical result for w for the case kH/Mg = 3.