Unformatted text preview: Dr. Galeazzi PHY205 Final Exam May 10, 2010 Signature: ___________________________ I.D. number: _________________________ Name:_______________________________ 1 You must do the first two problems which consists of five multiple choice questions each. 2 Then you must do four of the five long problems numbered 37. Clearly cross out the page and the numbered box of the problem omitted. Do not write in the other boxes. 3 If you do all the problems, only problems 16 will be graded. Each problem is worth 25 points for a total of 150 points. TO GET CREDIT IN PROBLEMS 3 7 YOU MUST SHOW GOOD WORK. 4 CHECK DISCUSSION SECTION ATTENDED: [ ] Dr. Nepomechie 2O, 9:30 10:20 a.m. [ ] Dr. Voss 2Q, 12:30 1:20 p.m. [ ] Dr. Voss 2R, 2:00 2:50 p.m. [ ] Dr. Huffenberger 2S, 3:30 4:20 p.m. [ ] Dr. Galeazzi 4P, 11:00 11:50 a.m. 7 6 5 TOTAL THE EQUATION SHEET IS PROVIDED ON THE LAST PAGE WHICH YOU CAN TEAR OFF. 1 Dr. Galeazzi [1.] PHY205 Final Exam May 10, 2010 This problem has five multiple choice questions. Circle the best answer in each case. [1A.] A geosynchronous satellite is transmitting the television signal for the Miami area. Which of Earth's big circles is the satellite following in its path? [d] The equator [1B.] Two cables, 1 & 2, are made of the same material, however, cable 2 is twice as long and has twice the diameter of cable 1. If they are stretched with the same force, cable 1 will stretch a length , while cable 2 will stretch a length . Which is true: . F and Y are the same, but cable 2 has twice the length and four time , or . the sectional area (twice the diameter), therefore [1C.] Particle A is moving with velocity velocity particle A. 3 9 7 4 , while particle B is moving with . Find the relative velocity of particle B with respect to [1D.] A car with mass M is going around a circular, flat curve with radius R. The maximum speed at which the car can travel without slipping is . What is the maximum speed at which a second car with mass 2M can travel on the same curve? Assume the coefficient of static friction is the same for both cars. Newton's first law: maximum speed is when ; . does not depend on m. Therefore the maximum speed is always [1E.] The position of a particle is described by the equation cos . What is the value of the phase if the particle is passing through the origin with positive velocity at time 0? From the initial condition at Moreover, 0 : @ : 0 2 Dr. Galeazzi [2.] PHY205 Final Exam May 10, 2010 This problem has five multiple choice questions. Circle the best answer in each case. with [2A.] You are crossing a river with width L where the river current has a velocity with respect to the river. In what respect to the ground, while your boat has a speed direction with respect to should you point your boat in order to reach the other side right across from your starting point? To land across the starting point the ycomponent of the velocity in the river banks reference frame must be 0. Therefore [2B.] If [2C.] A uniform meter stick with length L and mass M has a mass 2M attached to its left end, and another mass 3M attached to its right end. The stick is hanging from the ceiling through a line that is attached to the stick a distance x from its left end and it hangs horizontally in perfect equilibrium. What is the value of x? For equilibrium the hanging point must coincide with the CM of the whole system. Taking the origin at the left end: 3 4 6 and 5 2 3 , find . [2D.] A satellite flying on a circular orbit completes a full orbit every 100 minutes. What is the angular frequency of the satellite?
. . for [2E.] A disk with radius R has mass per unit area that depends on the radius as 0 , with positive and constant. What is the mass of the disk? 3 Dr. Galeazzi PHY205 Final Exam May 10, 2010 is rolling down an [3.] A hollow cylinder with mass M, radius R, and moment of inertia incline that forms an angle with the horizontal. The cylinder is released at rest at the top of the incline and rolls down the incline without slipping. The incline has total length L. Write your results in terms of M, R, L, , and g. Remember to check the units/dimensions. [a] Find the linear acceleration of the cylinder. [b] Find the position of the cylinder x as a function of time, where x is the distance from the top of the incline. [c] Find the speed of the cylinder at the bottom of the incline. [a] Using Newton's 2nd law for the x and y direction, plus the torque equation, plus the relation sin cos 0 0): for rolling without slipping (noting that 0, sin sin . 0 (take is as 0; [email protected] 0; 0): [b] The acceleration is constant, and the cylinder starts from rest at sin sin sin , where we used sin , where we used [email protected] [c] Cons. of Energy, using the flat surface as reference point: sin Also: Time for the cylinder to run a distance L: sin sin 2 4 Dr. Galeazzi PHY205 Final Exam sin 2 sin May 10, 2010 Speed at bottom is speed at : 5 Dr. Galeazzi [4.] The position expression With , , and PHY205 Final Exam May 10, 2010 of a particle with mass m as a function of time t is described by the 2 positive and constant. of the particle as a function of time t; of the particle with respect to the origin as a function of [a] Find the linear momentum [b] Find the angular momentum time t; [c] Find the net force acting on the particle as a function of time t; (and t). Remember to check the units/dimensions of the [d] Find the net torque with respect to the origin as a function of time t. Write your results in terms of m, , , results. [a] [b] 2 4 4 [c] [d] Also: 2 2 4
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2 4 4 2 4 8 0 0 4 12 20 4 4 4 2 4 4 4 4 4 4 12 2 4 4 2
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4 12 20 6 Dr. Galeazzi PHY205 Final Exam May 10, 2010 [5.] On a linear track a car (#1) with mass m moves with speed toward a second identical car (#2) which is at rest. The two cars are equipped with spring bumpers and the collision can be considered perfectly elastic. Assume there is no friction. [a] Derive the velocity [b] Derive the velocity of car #1 after the collision; of car #2 after the collision; [c] Derive the velocity of the center of mass of the two cars before the collision; [d] Derive the total work done on car #1 during the collision. Write your results in terms of m and , and MAKE SURE TO SHOW YOUR WORK. If you exclude any solution, explain the physical reasoning for that. Remember to check the units/dimensions for each answer.  2 0 2 2 0 #1 0 ; #2 We can exclude solution #2 as it represents the case where the first car misses the second and therefore there is no collision (the first car keeps moving at the same velocity, the second stays at rest). Therefore: [a] 0 [b] Check dimensions: [c] Before the collision: Check dimensions: [d] Work energy relation: OK OK 0 7 Dr. Galeazzi PHY205 Final Exam May 10, 2010 [6.] Two masses, and are connected to the ends of two strings of equal length d. The other ends of the strings are attached to the same nail in the ceiling (point P). Mass is initially raised, keeping the string straight, so that the string forms an angle with the vertical, while mass is just hanging from its string (see figure). When mass is released, it hits mass and sticks to it.
Derive: [a] the speed of mass just before it hits mass ; [b] the angular frequency after the collision; of the motion of the two masses of the motion after [c] the maximum angular displacement the collision. , , d, , and g. Remember to Write your results in terms of check the units/dimensions for each result. [a] Using point P for the origin and conservation of energy: cos 2 1 cos .
sin . [b] After the collision it is a simple pendulum with mass Assuming SHM: cos cos . Calculating the torque with respect to P when the mass forms an angle the only force contributing is the weight: Using the torque equation: . sin for 1. with the vertical, [c] First we find the speed of the two masses after the collision using conservation of momentum: 8 Dr. Galeazzi PHY205 Final Exam 2 1 cos . May 10, 2010 We then use conservation of energy to find the maximum angle: cos 2 cos 1 1 cos 1 cos cos . 9 Dr. Galeazzi PHY205 Final Exam May 10, 2010 [7.] Two particles with the same mass M are held in place on the xaxis on each side of the origin, both at a distance L from the origin. A third particle, with mass m, is free to move and is placed on the yaxis, a distance y from the origin. Assuming that the particles are in deep space, so that only the gravitation attraction between them is important: [a] Find the acceleration of the third particle; ); [b] Find its potential energy (take the energy to be zero at [c] Assume that the third mass is passing through the origin with initial velocity in the positive ydirection. Explain in less than 20 words why the mass will stay on the yaxis. (HINT: use the result from part [a] and what you know about linear motion). necessary for the third particle to escape the gravitation [d] Find the minimum value of attraction of the first two, i.e., the minimum value of to reach ). Write your results in terms of M, L, m, y, and G. Remember to check the units/dimensions of each result. [a] The third particle will feel the gravitational attraction of the first two. Because of the symmetry of the system, the xcomponents of the two forces cancel out (and the net xcomponent of the force will be zero), while the y components will add to each other. Therefore: 2 2 2 sin 2 [b] The potential energy is a scalar, so we simply add them: 2 2 2 . 10 Dr. Galeazzi PHY205 Final Exam May 10, 2010 [c] Initial position and velocity are aligned with the yaxis. The acceleration only has a ycomponent, therefore the mass cannot move from the yaxis. \ [d] To escape, the mass must reach 0 0 2 with at least zero speed. Cons. of energy: 0 . 11 Dr. Galeazzi PHY205 Final Exam
EQUATION SHEET (Make sure to start all problems from these equations) May 10, 2010 Vectors: 2D: cos , ; ; tan ; sin cos , sin . + R.H.R. for direction ; Linear motion: ; ; ; ; Relative velocity: Newton's Laws: Examples of forces: Work: Power: WorkEnergy Theorem: conservative forces: in general: examples: Force: Momentum and impulse: Center of mass: Circular motion: Uniform circular motion: ; Rolling without slipping: Moment of inertia: ; 0 ; ; ; 0; ; ; ; ; friction:  , 0   ; or ; ; ; or ; 2 ; , or  ; ; ;  ; ; ; ; ; Work and Energy of rigid bodies: Torque: Angular momentum: Simple Harmonic Motion: Young's Modulus: , or cos ; cos 12 ...
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 Galeazzi
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