Dr. Mills        sec 09833        Spring 2002

Answer all of the following questions for full credit on this exam.    Show all your work.    Circle the answer to each part of each problem you work.    Work the additional problem as an extra-credit problem.

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Part I: (Ch 1-9)      [grade for this section + Part II grade = Exam IV grade]

 

    1.    You are given two vectors: A = 5i + 10j - 3k and B = -2i + 5j + 2k.  Find (a) A+B, (b) A - B,     (c) AlB,    (d) A x B    (e) AXBlC   where C = 7i + 3j + 6k  (note thatl indicates a dot product)

    2.    A hot air balloon rises from the ground at the rate of 2.0 m/s.    A champagne bottle is opened to celebrate the event, expelling the cork with a speed of 5.0 m/s.    When opened, the bottle is pointed 20^o above the horizontal and is 6 m above the ground.

    (a)    What is the initial velocity of the cork as seen by an observer on the ground?

    (b)    How high does the cork go?

    (c)    How long is the cork in the air?    How long until it reaches its maximum height?

    (d)    How far (laterally) does the cork go?

    (e)    What is the velocity of the cork when it hits the ground?

 

    3.    A mass m is attached to mass M by a string and a pulley as shown in the figure. M is on the inclined plane. In part (a) consider the pulley as massless and frictionless and the coefficients of friction zero.    The mass M is 6.7 kg.    The angle the inclined plane makes with the horizontal is 40^o. Determine:

    (a)    the mass of m which produces an acceleration of mass M at 2m/s² up the plane.

    (b)    the tensions T₁ and T₂ for part (a)?

    (c)    the mass of m which produces an acceleration of mass m at 2m/s² upward if the coefficient of kinetic friction on the inclined plane is 0.1.

    (d)    the tensions T₁ and T₂ for part (c).

    (e)    the mass of m which produces an acceleration of mass m at 2m/s² downward if the coefficient of kinetic friction on the inclined plane is 0.1 and the pulley has a mass of 1 kg and a radius of 5 cm.

    (f)    the tensions T₁ and T₂ for part (e).                [30 points]

    4.    A 26,000 kg truck is moving at 10 m/s when it starts down a 5^o grade , from Donner Pass to Reno Nevada.    At the start of the descent the driver notes the altitude is 1650 m and the truck’s speed is 10 m/s.        When the driver reaches the foot of the grade, the altitude is noted to be 1450 m and the truck’s speed is 30 m/s.

    (a)    What is the change in the truck’s kinetic energy?

    (b)    What is the change in the truck’s potential energy?

    (c)    What is the amount of energy lost in applying the brakes, wind resistance, and other frictional losses during the descent?

    (d)    if wind resistance provides a retarding force proportional to the velocity of the box where the constant of proportionality, k, is 500 kg/s (i.e. air friction is = kv), what is the amount of energy dissipated in wind resistance?  (assume that the truck’s speed constantly increases over the 2295 m distance)
 
 

    5.    A metal block of mass m₁ is attached to a spring which in turn is attached to the ceiling.    A string from mass m₁ supports a second block of mass m₂ which is suspended from the first block.    The system is in static equilibrium, then the string is cut!

    (a)    What is the net force acting on the two-block system before the string is cut? (draw a force diagram for each mass).

    (b)    What is the net force acting on the two-block system after the string is cut?    (draw a force diagram for each mass).

    (c)    What is the acceleration of the center of mass of the two-block system immediately after the string is cut?

    (d)    What is the acceleration of each mass system immediately after the string is cut?

    (e)    Describe what happens to each mass!

 

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 Part II:    (Ch 10-12)     [Exam III grade for this section]     Work all 5 problems!

    6.    The Bohr model of the hydrogen atom pictures the electron as a tiny particle moving in a circular orbit about a stationary proton.    In the lowest energy orbit the distance from the proton to the electron is 5.29 x 10^-11 m, and the linear speed of the electron is 2.18 x 10⁶ m/s.

    (a)    What is the angular velocity of the electron (don’t forget to tell me the direction)?

    (b)    How many orbits about the proton does it make each second?

    (c)    What is the electron’s centripetal acceleration?

    (d)    What is the electron’s angular momentum?    (10.35)

    7.    A 2.0 kg cylinder of radius 0.1 m and length 0.5 m rolls from rest down a ramp (without slipping).    The ramp is 0.75 m high and 5.0 m long.    When the cylinder reaches the bottom of the ramp, what is:                

    (a)    its rotational kinetic energy?

    (b)    Its translational kinetic energy

    (c)    Its total kinetic energy

    (d)    Its angular velocity

    (e)    Its acceleration                                                    [30 points]

    8.    An 85 kg person stands on a ladder which is 8 m long and weighs 98 N.   The angle between the ladder and the floor is a 60 degree angle.    The floor is rough with a coefficient of friction of 0.1.    Treat the wall as if it is frictionless.    Calculate:

        (a) how high up the ladder the person can climb before the ladder starts to slip,

        (b) the normal force of the ladder against the floor and the wall for (a),

        (c) the frictional force of the ladder against the floor,

        (d) the torque produced by the person on the ladder calculated about the ladder’s base.

    9.    A rigid vertical rod of mass M rests on the floor and is held in place by static friction.    It is held in place by a wire connected between its top end and the floor (and makes an angle of 45^o at the floor).    If a horizontal force F is applied at the midpoint between the floor and the wire pushing the rod away from where the wire is attached to the floor, what is the maximum force F that can be applied before the rod slips?        

    10.    The Martian moon Deimos (16 km long) has a period that is greater than the other Martian moon, Phobos (27 km long).    Both moons have approximately circular orbits.

        (a) Is Deimos closer to or farther from Mars than Phobos?    Explain.

        (b) Calculate the distance from the center of Mars to Deimos given that its period is 1.10 x 10⁵ s.

        (c) If Phobos is 9,380 km from the center of Mars – what is its period?

        (d) If Mars rotates once every 24 hours and 37 minutes, which way would you see each moon pass and how many times a day would you see each moon if you were standing on Mars?

 

g is the acceleration of gravity on the earth = (9.8 m/s²)                        G = 6.67 x 10^-11 Nm²/kg²