Phys 250 Midterm Review Spring 2016 Name: 1. Consider a simple pendulum with a mass of 10 kg at the end and a length of 5 meters. A: If the pendulum is given 10 J of KE at the equilibrium point, what would be the velocity of the weight? $\approx 1.4 \pu{ ms^{-1} }$ B: If there is a fluid surrounding the pendulum that acts to damp out the oscillations with a damping coefficient of B=0.01 N/(m/s), how long until the velocity of the weight at equilibrium will be 1/30th of the initial velocity? $\huge $ 2: Consider a water tower that is a cylinder 15 meters across and 30 meters tall and is completely filled with water of density 1000 kg/m^3.   A: What is the difference in pressure from the top of the cylinder to the bottom? B: Ignoring air pressure, if a spigot is opened at the bottom of the cylinder, at what velocity will the fluid be ejected out according to Bernouli’s principle? 3: Consider a coupled harmonic oscillator of spring constant k=25 N/m and two masses such that M1 = 5 kg and M2=2 kg. A: What is the reduced mass MR of this system? B: What is the natural frequency of oscillation for this system? C: If we stretch the spring so that there is an initial displacement of 1 meter from equilibrium at t=0, then what are the maximum velocities reached by M1 and M2? 4: A pendulum on Mars is 5 meters long, and at the end has a 10 kg weight.  The Martian gravity is 3.7 m/s2. A: Treating the pendulum as a simple harmonic oscillator, what is the natural angular frequency ω of the pendulum? B: The Russian Space Agency gives the pendulum 4 J of kinetic energy.  Keeping with the simple harmonic oscillator approximation, what will the maximum displacement of the pendulum be from equilibrium? C: What is the maximum velocity reached by the pendulum weight in m/s? 5: During a hard rain in downtown Seattle, the water flows down the hill on Madison street.  The water is near the surface of the Earth, and has a density of 1000 kg/m^3. A: If the water begins flowing at 0.5 m/s, what is the kinetic energy density of the water? B: After the water has flowed down the hill and dropped 150 m in altitude, what is the new velocity of the water? 6: A city reservoir is 20 meters deep filled with water that has a density of 1000 kg/m^3.  A sphere of lead has a radius of 0.5 m and a density of 11,300 kg/m^3 when it’s dropped into the pool. A: What is the buoyant force on the sphere when it’s resting on the bottom? B: If you fill a raft with carbon dioxide with a density of 1.98 kg/m^3, what volume of raft do you need to place below the sphere to completely lift the sphere out of the water?  Neglect the mass of the raft material. 7:  You are presented with a hanging spring with a weight on the end that is oscillating up and down as a simple harmonic oscillator, near the surface of the Earth.  You observe that the velocity of the spring systems follows a function of v(t) = 13 cos(4.5*t + 5) meters/second. A: If the weight has a mass of 2 kg, what is the spring constant k? B: What is the maximum KE of the system and what is the maximum PEspring of the system? C: What was the initial KE, PEspring, and PEgrav of the system, if we define PEgrav=0 when x=0? 8: A 200 gram bullet is fired at 250 m/s into the weight of a simple pendulum 4 meters long hanging at rest. A: If the pendulum weight began with a mass of 10 kg and the bullet collided with it inelastically and is now stuck in the weight, what is the initial velocity of the pendulum weight? B: If the damping coefficient of the pendulum interacting with the air around it is 0.1 Ns/m, show that this is an underdamped pendulum and write an equation of motion for the velocity of the damped pendulum with time given the information in the problem.   C: How long will it take the pendulum to decrease to 1/10th of its initial velocity? 9: A stone chest is flooded with water at the bottom of the meditteranean.  The stone has a density of 6000 kg/m^3 A: If lid of the chest is 2.5 m x 1m in area and 4 cm thick, what is the mass of the lid? B: How fast does the water have to flow over the top of the chest before the lid will lift off of it? 10: Consider an instructor with a mass of 115 kg trapped in a rectangular coffin, 2.5m x 1m x 0.5m, with the rest of the volume of the coffin filled with air (neglect the mass of the coffin).  A: If the coffin is floating on the surface of a body of water with density equal to 1000 kg/m^3, what faction of the volume of the coffin will be submerged? B: If instead the coffin is floating a lake of liquid mercury with density equal to 13,500 kg/m^3, what fraction will be submerged?