Final Exam Spring 2014 May 05, 2014

Transcription

1 Final Exam Spring 2014 May 05, 2014 Section number Section instructor Last/First name Last 3 Digits of Student ID Number: Answer all questions, beginning each new question in the space provided. Show all work. Show all formulas used for each problem prior to substitution of numbers. Label diagrams and include appropriate units for your answers. Write your name and section number at the top of each page in the space provided and write the name of your section instructor in the place provided in the cover sheet.you may use an alphanumeric calculator (one which exhibits physical formulas) during the exam as long as you do not program any formulas into memory. By using an alphanumeric calculator you agree to allow us to check its memory during the exam. Simple scientific calculators are always OK! A Formula Sheet Is Attached To The Back Of This Examination For your convenience you may carefully remove it from the Exam. Please take it with you at the end of the exam or throw it in a waste basket. Be Prepared to Show your Student ID Card Score on each problem: 1. (25) 2. (25) 3. (25) 4. (25) 5. (25) 6. (25) 7. (25) 8. (25) Total Score (out of 200 pts)

2 1. (25 point) Put a circle around the letter that you think is the best answer (5 pts) If a particle undergoes SHM with amplitude 0.15 m, what is the net displacement of the mass after a time interval T? A) 0 m B) 0.30 m C) 0.45m D) 0.60 m E) none of the above (5 pts) A mass m is attached to a spring with spring constant k. When this system is set in motion with amplitude A, it has a period T. What is the period if the amplitude of the motion is increased to 2A? A) 2T B) T/2 C) T. D) 4T E) T 1.3. (5 pts) A merry-go-round spins freely when Janice moves quickly to the center along a radius of the merry-go-round. It is true to say that A) the moment of inertia of the system decreases and the angular speed increases B) the moment of inertia of the system decreases and the angular speed decreases C) the moment of inertia of the system decreases and the angular speed remains the same. D) the moment of inertia of the system increases and the angular speed increases. E) the moment of inertia of the system increases and the angular speed decreases.

3 (5 pts) When you ride a bicycle, in what direction is the angular velocity of the wheels? A) to your left B) to your right C) forwards D) backwards E) up 1.5. (5 pts) If the net work done on an object is negative, then the object's kinetic energy A) decreases B) remains the same C) increases D) is zero E) cannot be determined without knowing the object's mass

4 4 2. a) (12 pts) What is the maximum speed at which a car can round a curve of 20 m radius on a level road if the coefficient of static friction between the tires and road is 0.60? b) (13 pts) Two objects are moving as shown in the figure. What is the total angular momentum about point O?

5 3. (25 pts)a vertical spring (ignore its mass), whose spring constant is k=875 N/m is attached to a table and is compressed down by m. (a) (13 pts) What upward speed can it give to a m=0.380 kg ball when released? 5 (b) (12 pts) How high above its original position (spring compressed) will the ball fly?

6 4. (25 pts) A m=2.0 kg block is attached to a massless string that is wrapped around a M=1.0 kg, R=20.0 cm radius cylinder, as shown in the figure. The cylinder rotates on an axel through the center. The block is released from the rest h=1.0 m above the floor. The moment of inertia of the cylinder is I = 1 2 MR2 a) (4 pts) Draw a free-body diagram. b) (14 pts) Find acceleration of the block and the tension in the wire. 6 c) (7 pts) How long does it take for the block to reach the floor?

7 7 5. (25 pts) In the figure shown below, a small body of mass m=1.00 kg swings in a vertical circle at the end of a cord of length R=1.0 m. The speed v = 2. 0 m/s when the cord makes an angle θ = 33 with the vertical. a) (4 pts) Draw a free-body diagram and resolve forces into radial and tangential components; b) (4 pts) Find the radial acceleration; c) (5 pts) Find the tangential acceleration; d) (5 pts) Find the magnitude and direction of the resultant acceleration; e) (7 pts) Find the tension in the cord.

8 8 6. (25 pts) A m=500 g block and a spring are arranged on a horizontal, frictionless surface. Position-versus-time graph for the oscillator is shown in the graph. The oscillations are measured to have a period of 2.0 s. a) (7 pts) Write down the amplitude (A), angular frequency (ω), and phase angle (φ) b) (2 pts) Write down a function which describes this simple harmonic motion c) (3 pts)what is the spring constant, k? d) (3 pts)what is the maximum speed of the block? e) (3 pts) Write a velocity as a function of time. f) (7 pts) At what position is the block s speed v=1.0 cm/s

9 9 7. (25 pts) A bullet of mass m moving with velocity v strikes and becomes embedded at the edge of a cylinder of mass M and radius R 0. The cylinder, initially at rest, begins to rotate about its symmetry axis, which remains fixed in position. Assume there is no frictional torque. a) (5 pts)what is conserved during this collision? Why is it conserved? b) (20 pts) What is the angular velocity of the cylinder after this collision?

10 8. (25 pts) An m=80 kg worker sits down 2.0 m from the end of an M=1450 kg steel beam of length l = 6. 0 m to eat his lunch. The cable supporting the beam is rated at F TTTT = 1111 N. a) (5 pts) Draw a free-body diagram of the beam b) (10 pts) Find the tension in the cord, F T. Should the worker be worried? c) (10 pts) Find the horizontal and vertical forces exerted by the wall on the beam. 10

11 Physics I Formula Sheet Translational Motion x= x 2 x 1 (displacement) v average = x/ t a average = v/ t Given x(t) v(t) = dx/dt a(t) = dv/dt = d 2 x/dt 2 Kinematic eq-ns with const. Acc.: v(t) = v 0x +at x(t) = x 0 + v 0x t +(1/2) at 2 v 2 = v 0x 2 + 2a(x x 0 ) Newton 2 nd law F = ma F = dp dd Frictional Forces: Fs µsf N Fk = µ k F N For springs: F = -kx U(x) = (1/2)kx 2 Linear Momentum and Impulse d d d J = Fdt = Fav t For elastic collision: For 1-D elastic head-on collisions: v A vb = ( v' A v' B ) Work and Kinetic Energy W = Flcosθ r 2 W = F dr r 1 K trans = (1/2)mv 2 ; K rot = (1/2)Iω 2 K tot = (1/2)I CM ω (1/2)Mv CM Work-Kinetic Energy principle Wnet = K With non-conservative forces: K + U = W NC Centripetal acceleration: a R = v 2 /R; a R = ω 2 R Rotational Motion θ= θ 2 θ 1 ω = dθ/dt α = dω/dt Given θ(t) ω(t) = d θ/dt α(t) = dω/dt = d 2 θ/dt 2 Rotat. kinematic eq-ns with const. angular acceleration ω(t) = ω 0 +αt θ(t) =θ 0 +ω 0 t +(1/2)αt 2 ω 2 = ω α (θ θ 0 ) Rotat. Newton 2 nd law τ = Iα τ = dl dd Angular Momentum L = r p ; L = I ω I = ΣmiRi 2 Torque τ = r F ; τ = r F sinθ Potential Energy U = U ( x) U 0 ( x0 ) F(x) = - du(x)/dx For gravity on earth's surface: F g = mg U(y) = mgy For gravity in general: F g = - GmM E /R 2 U(r) = - GmM E /R g = 9.8 m/s 2 ; G = 6.67 x N.m 2 /kg 2 Total mechanical energy: E tot = K + U Power P avg = W/t; = P = dw/dt; Equations connect. trans./rotat. motion v tan = Rω a tan = Rα x x0 Fdx

12 12 Center of Mass r cm = Σm i r i /M ΣF ext = Ma cm Differentiation: dx n /dx = nx n-1 (n 0) dcos(x)/dx = -sin(x) (x in radians) dsin(x)/dx = cos(x) (x in radians) Right triangle: sin θ = a/c cos θ = b/c tan θ = a/b c 2 = a 2 + b 2 Quadratic Formula: Ax 2 + Bx + C = 0 has solutions: B ± B 2 4AC x = 2A Harmonic Motion F = kx U(x) = (1/2) kx 2 ω = k ; T = 2π m T = 2π/ω ; f =1/T ; ω = 2πf x(t) = Acos(ωt + φ) v(t) = ωasin(ωt + φ) a(t) = ω 2 Acos(ωt + φ) Simple Pendulum T = 2π L ; ω = g g L Kepler's third law: T 2 /R 3 = 4π 2 /GM sun Integration: n+ 1 n x x dx = + C n + 1