PHY-251 Final Fall 2020

Name:

Multiple Choice Clearly circle the correct answer. Show your work in the open space to be considered for partial credit.

1. Two balls are thrown reach the same height. Which of the following is definitely true? a) The x-components of the initial velocity are the same. b) The y-components of the initial velocity are the same. c) Both balls land at the same horizontal distance. d) The magnitudes of the initial velocity are the same.

2. A rollercoaster of mass m starts from rest at height y1, however because the track was not properly greased only makes it to the height y2 before it comes to rest (there are safety brakes that keep the coaster from rolling backwards). The work done by friction = a) mg(y2 ? y1) b) µKmg(y2 + y1) c) -mg(y2 ? y1) d) -µKmg(y2 ? y1) e) Not enough information to tell.

Figure 1: Figure 2:

3. Figure 2 represents what type of phenomena? a) sheer stress/strain. d) thermal stress/strain. b) tensile stress/strain. e) bulk stress/strain. c) squishing.

4. A cylindrical aluminum can of radius R=9.5 mm, L=81 cm, and modulus G=2.6×1010N/m2 has a F=63 kN force applied as in Figure 2. Calculate the strain ?x/L.

5. True or False. Conservative forces do not depend on path.

6. Vectors ~r1 = – 3.0 m j? and ~r2 = 3.0 m i? describe an object’s position. Draw vector ?~r = ~r2 – ~r1 AND calculate the magnitude ?r and direction ?, of ?~r.

7. An object rotates initially at ?0=13.0 rotations per minute (rpm) and increases to ?=21.0 rpm in t=60 s. Calculate the angular acceleration of the object.

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PHY-251 Final Fall 2020

Directions: Please show all work. Box and clearly label your final answers.

Figure 3: Figure 4:

Question 1. You throw a ball toward a wall from an initial height y0=1.4 m at speed v0=25.0 m/s and at angle ?=40.0? above the horizontal. The wall is distance d=2.0 m from the release point of the ball.

A. Calculate the height of the ball when it hits the wall.

B. Calculate the magnitude of the velocity when it hits the wall. (Hint: You may use kinematics OR conservation of energy).

Bonus: When it hits the wall, has it passed the highest point on its trajectory? Show quantitatively.

Question 2. A cart of mass m1=5.0 kg travels at vi1=6.28 m/s toward a cart of mass m2=3m1 that is at rest as in Figure 4. Momentum is conserved and the collision is completely inelastic, so the two masses travel together with final velocity vf . What is the final velocity vf?

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PHY-251 Final Fall 2020

Figure 5:

Question 3. A solid cylinder of mass M=5 kg and radius R=0.1 m rolls without slipping with velocity V=8.4 m/s toward an incline defined by angle ?=10? as in Figure 5. (Icyl=

1 2MR

2)

A. Calculate the kinetic energy of the sphere right before it goes up the incline.

B. Calculate the height the mass gets up the ramp.

C. Consider a box with the same mass and velocity instead sliding without friction toward the incline. How does this object’s kinetic energy before the incline compare to the sphere?

D. How high up the ramp does the box get? How do the heights compare between the objects? Explain.

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PHY-251 Final Fall 2020

Figure 6:

Question 4. Three cylinders initially rotate about a massless rod without making contact as in Figure 6. The yellow cylinder has moment of inertia Iyellow=0.5 kg m

2 and rotates counter-clockwise (C.C.W.) with ?yellow= 3.0 rad/s, the blue cylinder has Iblue=4.0 kg m

2 and rotates clockwise (C.W.) with ?blue= 1.5 rad/s, and the red cylinder has Ired=13.5 kg m

2 and rotates counter-clockwise (C.C.W.) with ?red= 1.0 rad/s. The discs then drop and are allowed to rotate together as a system reaching the same final angular speed, ?f .

A. Calculate the moment of inertia of the system about the rotation axis after the discs make contact.

B. Calculate the final angular velocity ?f .

Bonus. Is (rotational) kinetic energy conserved in this process?

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PHY-251 Final Fall 2020

Figure 7: Figure 8:

Question 5. A massless rope is wrapped around the center of a solid sphere of mass Msphere=5.0 kg and radius R = 0.20 m that is fixed and allowed to spin about its center of mass. The rope is connected to a mhanging=5.0 kg hanging mass as in Figure 7. The mass falling causes the sphere to rotate without slipping. Hint: Think of a pulley system where the pulley is a solid sphere Isphere=

1 2MR

2.

A. Write Newton’s 2nd Law for the hanging mass.

B. Write Newton’s 2nd Law (rotational) for the sphere.

C. Calculate the acceleration of the mass.

Question 6. A Mhang=5.0 kg mass hangs from the end of a L=3 m and Mrod=2Mhang=10.0 kg. At what distance D (as drawn in Figure 8) can the rod be balanced such that the system stays in equilibrium (i.e. ?=0 and a=0)?

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PHY-251 Final Fall 2020

Figure 9:

Question 7. A mass m=680 g and spring with spring constant k oscillates with angular frequency ?=9.8 rad/s, position amplitude xm=11 cm, and phase ?=0 as in Figure 9.

A. Calculate the spring constant k.

B. Calculate the maximum acceleration.

C. At what position(s) does the mass reach its maximum velocity? Explain.

D. Calculate the velocity at t=3.0 s.

E. Calculate the total mechanical energy E of the system?

Bonus. Calculate the kinetic energy K of the system at t=3.0 s.

Answer EITHER 8A or 8B. Indicate clearly on your test which you are solving for credit, otherwise you will lose credit. You will only be given credit for the problem you choose.

Question 8. Venus is approximately aVenus=0.7 AU (where 1 AU=aEarth=1.496× 1011 m) from the Sun in its orbit. The gravitational constant G=6.67×10?11 N m2/kg2, MSun=1.989×1030 kg, and MVenus=4.867×1024 kg.

A. Calculate the gravitational force between Venus and the Sun.

B. Calculate the period of Venus around the Sun (in seconds or years).

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PHY-251 Final Fall 2020

Answer EITHER 9A or 9B. Indicate clearly on your test which you are solving for credit, otherwise you will lose credit. You will only be given credit for the problem you choose. Each problem has a bonus option, however there are no bonus points awarded to solving both 9A and 9B.

Figure 10: Figure 11:

Question 9A. A pendulum of mass M = 135 g and period T=1.64 s as in Figure 10. What is the length L of the pendulum?

Bonus. If the initial angle ?=10?, what is the kinetic energy of the mass at the lowest point in it’s motion?

Question 9B. A beam of mass M=5 kg and length L=1 m is fixed with a hinge and cable at ?=30? as in Figure 11. Calculate the tension.

Bonus. The cable breaks and the beam is able to rotate about the hinge. Calculate the angular acceleration of the beam (take the moment of inertia about the hinge to be I = 13ML

2).