Practice Problems for Final Exam

  1. Artillery Captain Smith sets up his big gun on a flat plain, just below a steep hill. His gun fires shells with a muzzle velocity of v = 800 m/s. Smith sets the elevation angle of the gun to θ = 30 degrees, and aims toward enemy lines, which are far away across the long, flat landscape.

    1. How far will the shells travel?
    2. How long will they spend in the air?

    "This won't work," decides Smith. "We can't see the enemy positions clearly from this location. Let's move the gun up to the top of that nearby hill, which is H = 1300 m tall."

    1. When launched from this position, how far will the shells travel horizontally?
    2. Bonus! If Smith changes the elevation angle of his gun, he can increase its range. Which angle will yield the largest range?

    The detailed solution:

  2. A circular disk of radius R is made of material which is very dense around the edges, but not very dense near the center. The surface density (mass per unit area) at a distance r from the center is described by the equation 
                                             (       r  )
            surface mass density =   K * (  1 + --- )
                                         (       R  )
    where K is some constant with units of kilograms per square meter.

    1. What is the mass of this disk, in terms of K and R?
    2. What is the moment of inertia of this disk around its center?

      3. The disk is put on a ramp which is tilted at angle theta above the horizontal. The starting point of the disk is a vertical height H above the bottom of the ramp. The disk is held motionless, then released. It starts to roll without slipping.

    3. How long will it take the disk to roll down to the bottom of the ramp?

    The detailed solution:

  3. The amusement park builds a big wave tank. The waves are created by a simple machine: a big panel which is pushed forward by 0.8 m, the back again, repeatedly, every 1.6 s. The wave crests move through the tank with a speed of 60 m/s. One particular crest appears at x = 5 m and t = 2 s.

    1. Write an equation which expresses the height of the surface of the water as a function of position and time: 
                        y(x,t) = sin(kx - wt + phi)
      Provide values for k, w, phi.
    2. The foreman realizes that he could increase the height of the waves by making the length of the tank a particular length. What length will do the trick?

    The detailed solution:

  4. Little Billy (m = 30 kg) stands at the edge of the merry-go-round platform. The platform is a disk of radius of R = 3 m and a mass of M = 180 kg. The platform is initially at rest.

    Billy throws a baseball (mb = 0.14 kg) tangent to the disk at a speed of v = 20 m/s.

    1. What is the angular velocity of the merry-go-round after he throws the ball?

    The detailed solution:

  5. The old cider mill has a waterwheel with paddles that extend H = 3 m below the wheel's central axle. A creek runs due North-east past the old mill. As the water flows by the mill, it exerts a force F = 20 N on the paddles.

    Express all quantities in unit-vector notation, using a coordinate system centered on the axle of the wheel with directions

    1. What is the position of the paddle in the water?
    2. What is the force of the water on the paddle?
    3. What is the torque due to this force?
    4. The moment of inertia of the wheel is I = 500 kg-m^2. If the wheel is initially at rest at time t = 0, what is its angular velocity one minute later?

    The detailed solution: