Letter. (A) Show that the following relationship must hold among V 0, R, and 0. for the marble to slide to a complete stop: (3 pts) 2 V=0

Size: px

Start display at page:

Download "Letter. (A) Show that the following relationship must hold among V 0, R, and 0. for the marble to slide to a complete stop: (3 pts) 2 V=0"

Harriet Hill
5 years ago
Views:

1 Letter Comprehensive Exam Session III Classical echanics Physics Department- Proctor: Dr. Drake itchell (Sun. Jan 12 th, 2014) (3 hours long 1:00 to 4:00 P) If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Express all the answers in terms of the given symbols for the various physical quantities unless otherwise stated differently in the problem. Show steps clearly. 1. An amusing trick is to press a finger down on a marble (taken as a uniform solid sphere of mass and radius R), on a horizontal table top, in such a way that the marble is projected along the table with an initial linear speed V 0 and an initial backward rotational speed of 0, 0 being about a horizontal axis perpendicular to V 0. The coefficient of sliding friction between the marble and table top is constant. Given: g = acceleration due to gravity, assumed constant oment of inertia for a solid sphere about an axis through its center = 2 R 2, oment of inertia for a thin rod about an axis through one of its ends = 1 L 2, 3 where R and L are the radius of the sphere and length of the rod, respectively. 5 (A) Show that the following relationship must hold among V 0, R, and 0 for the marble to slide to a complete stop: (3 pts) 2 o V0 0R R V o 5 x V=0 (B) Now if we desire the marble to first skid to a stop and then start returning toward its initial 3 position, with a final constant linear speed of V 0 ; show that a new relationship must 7 then hold among V 0, R, and 0 which is given by the following: (7 pts) 1 V0 0R o 4 3/7V o x V=0

2 2. A thin uniform rod AB of mass and length L is free to rotate in a vertical plane about a horizontal axis at end A. A piece of putty, also of mass, is thrown with velocity V horizontally at the lower end B while the bar is at rest. The putty sticks to the bar. What is the minimum velocity of the putty before impact that will make the bar rotate all the way around A? (10 pts) A g L V B

3 3. Lagrangian Consider a pendulum of mass m and length l, which is attached (via a light and inextensible string) to a support with mass, which can move along a line in the x-direction. Let x be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle θ from the vertical. (A) Write down the Lagrangian of the system and m in terms of the generalized coordinates and velocities. (2 pts) (B) Obtain the two Euler-Lagrange equations of motion to arrive at a system of coupled differential equations in terms of the two generalized coordinates. (4 pts) (C) For small oscillations of m about the vertical through, show that the period of the pendulum is approximately given by the following:. T 2 ( m ) g Comment on the result as the mass gets much larger than m. (4 pts).

4 4. Rocket motion Consider a rocket loaded with fuel in free space (no external forces are acting) initially at rest. The initial mass of the rocket + fuel system is kg. The mass of the fuel by itself is kg. The rocket moves forward by the reaction of ejecting mass (i.e. burning fuel) at high velocities. Assume that the rocket burns fuel (i.e. ejects mass) at a constant rate and the exhaust velocity of the fuel relative to the rocket is constant at 2600 m/s. Find the final velocity of the rocket after the fuel runs out. [Hint 1: Apply the law of conservation of linear momentum and solve the resulting equation for the rocket velocity v. Hint 2: You can neglect the product of two differentials.]

5 5. Terminal Velocity (A) A skydiver jumps off a tall building. Assume the force of air resistance on the skydiver is given by kmv, where k is a constant, m is the skydiver s mass, and v is the skydiver s velocity. Derive an equation for the velocity of the skydiver as a function of time. (B). Assume the building is tall enough so the skydiver reaches terminal velocity. Using your result from part (A) show the terminal velocity of the skydiver is g/k (assuming down is positive), where g is the acceleration due to gravity. (C) Graph the velocity of the skydiver up until the time she reaches terminal velocity. [Hint: If you do not know the antiderivative in part (A) you can find it by the substitution method.]

EXAM 3 MECHANICS 40% of the final grade

EXAM 3 MECHANICS 40% of the final grade Winter 2018 Name: Each multiple-choice question is worth 2 marks. 1. The mass of the two wheels shown in the diagram is the same. A force of 1 N is exerted on the