1. A mechanical system with one degree of freedom oscillates about a stable equilibrium state. Its displacement from the equilibrium, x(t), satisfies the simple harmonics oscillator equation:

d2x + ω2 x = 0.dt2

(a) What is the characteristic period of the oscillation?
(b) Write a solution to the above equation for which x(0) = x0 and ẋ(0) = v0. (c) Demonstrate that E = ẋ 2 + ω 2 x 2 does not vary in time. What is the physical significance of E?

2. An damped mechanical system with one degree of freedom oscillates about a stable equilibrium state. Its displacement from the equilibrium, x(t), satisfies the damped harmonic oscillator equation:

ẍ+νẋ+ω2 x=0, where ν > 0.

(a) Demonstrate that

x = A sin(ω1 t)eγt

is a particular solution of the damped harmonic oscillator equation, and determine the values of ω1 and γ (assuming that ν < 2 ω).

(b) Demonstrate that

dE ≤ 0,dt

where E is defined in Q1(c). What is the physical significance of this equation?

3. Consider a mass-spring system consisting of two identical masses, of mass m, which slid over a frictionless horizontal surface. In order, from the left to the right, the system consists of a spring of spring constant k′ whose left end is attached to an immovable wall and whose right end is attached to the first mass, a spring of spring constant k whose left end is attached to the first mass and whose right end is attached to the second mass, and a spring of spring constant k′ whose left end is attached to the second mass and whose right end is attached to an immovable wall. Let ω0 = √k/m and α = k′/k.

(a) Demonstrate that the equations of motion of the system can be written:

ẍ1 = −(1+α)ω02 x1 +ω02 x2,

ẍ2 = ω02 x1 −(1+α)ω02 x2,

(b) Demonstrate that the normal frequencies of the system are ω = α1/2 ω0 and ω = (2 +α)1/2 ω0.

(c) Demonstrate that the low frequency normal mode is such that the two masses oscillate in phase with the same amplitude, and that the high frequency mode is such that the two masses oscillate in anti-phase with the same amplitude.

The wheel of an engine starts to rotate from rest with uniform angular acceleration. It accelerates angularly for an unknown time (t1) until reaching an angular velocity of 45.13 rad / s, then maintains that angular velocity constant for 37.8 s. The total angle traveled by the wheel in total is 2398.61 rad.

a) What is the angular acceleration with which the wheel began to rotate?
b) What is the time (t1) that the wheel was accelerating?
c) What is the angle traveled by the wheel during acceleration?
d) How many laps did the wheel (revolutions) give? rev

You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isnt flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m/s. Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 8.20 m/s . You travel due south from one bank to the other and cross the river in 11.2 s.

With the throttle set so that the speed of the boat relative to the water is 6.00m/s, what is the shortest time in which you could cross the river?

A mass of 3.8 kg is originally moving at 8 m/s at the top of a frictionless incline which has a length of 6.2 meters and an inclination angle of 56 degrees. It slides down the incline and over a horizontal surface in which a portion of it has friction. The coefficient of kinetic friction is 0.37 and the portion of the surface which has friction is 8 meters. At the end of the horizontal surface is a spring. The mass compresses the spring 0.65 meters before it is stopped. What is the amount of force produced by the spring when the mass is stopped in Newtons?

A 55.0 kg box hangs from a rope. What is the tension in the rope if: Part A The box is at rest? Express your answer with the appropriate units. Part B The box moves up a steady 4.90 m/s ? Express your answer with the appropriate units. Part C The box has vy = 4.60 m/s and is speeding up at 4.90 m/s2 ? The y axis points upward. Express your answer with the appropriate units. Part D The box has vy = 4.60 m/s and is slowing down at 4.90 m/s2 ? Express your answer with the appropriate units.

In a capacitor charging and discharging experiment with a resistor of 10 ohms, if we plot our data so LVC (volts) is in the y-axis and time (seconds) in the x-axis, what would the slope of the resulting line represent? how do we use this slope to calculate the capacitance of this capacitor?

When a constant force acts on an object, what does the object's change in momentum depend upon?

You have been called to testify as an expert witness in a trial involving a head-on collision. Car A weighs 1515 lb and was traveling eastward. Car B weighs 1125 lb and was traveling westward at 41.0 mph. The cars locked bumpers and slid eastward with their wheels locked for 18.5 ft before stopping. You have measured the coefficient of kinetic friction between the tires and the pavement to be 0.750 . How fast (in miles per hour) was car A traveling just before the collision? (This problem uses English units because they would be used in a U.S. legal proceeding.)

Please show work!

a) A 80 kg baseball player jumps with a horizontal velocity of 8.1 m/s and collides with an 89

kg teammate who jumps with a horizontal velocity of 2.7 m/s in the opposite direction. If the players became tangled in mid-air, what would their combined horizontal velocity be after the impact?

A) 2.41 m/s   B) 2.99 m/s   C) 5.26 m/s   D) 5.54 m/s

b) After the collision, which horizontal direction do the tangled players move?

A) the players do not move after the collision   B) in the initial direction of the 80 kg player C) in the initial direction of the 89 kg player D) the players move vertically upward after the collision

c) A baseball (weight 5 oz.) thrown at 88 mph strikes a 30 oz. bat moving at a velocity of 72 mph. The coefficient of restitution between the baseball and bat is 0.49. How fast is the bat moving after it strikes the baseball?

A)   37.9 mph B) 48.2 mph C) 60.3 mph D) 63.1 mph

d) How fast is the baseball moving after striking the bat in part c)?

A) 54.6 mph   B) 116.3 mph C) 138.7 mph D) 141.5 mph

e) If the coefficient of restitution was higher in part c), how would that affect the velocity of the ball after the collision (no calculations needed)?

A) the final velocity of the ball would be zero B) the final velocity of the ball would be slower C) the final velocity of the ball would not change D) the final velocity of the ball would be faster

A skydiver jumps out of an airplane at a high altitude and reaches terminal speed.

Which of the following statements about the falling skydiver at terminal speed is FALSE?

1) The gravitational potential energy of the skydiver-Earth-atmosphere system is decreasing.

2) The thermal energy of the skydiver-Earth-atmosphere system is increasing.

3) The air resistance force does negative work on the skydiver.

4) At terminal speed, the kinetic energy of the skydiver is constant.

5) The gravitational force on the skydiver must be larger than the air resistance force in order to maintain a downward velocity.

The potential energy stored in the compressed spring of a dart gun, with a spring constant of 36.00 N/m, is 1.440 J. Find by how much is the spring is compressed. A 0.070 kg dart is fired straight up. Find the vertical distance the dart travels from its position when the spring is compressed to its highest position. The same dart is now fired horizontally from a height of 4.30 m. The dart remains in contact until the spring reaches its equilibrium position. Find the horizontal velocity of the dart at that time. Find the horizontal distance from the equilibrium position at which the dart hits the ground.

A soccer ball is kicked from the top of one building with a height of H₁ = 30.1 m to another building with a height of H₂ = 10.8 m. (It is not a very smart idea to play soccer on the roof of tall buildings.)

The ball is kicked with a speed of v₀ = 19.3 m/s at an angle of θ = 74.0° with respect to the horizontal. The mass of a size 5 soccer ball is m = 450 g. What is the speed of the soccer ball, when it lands on the roof of the second building? The soccer ball is kicked without a spin. Neglect air resistance.

An unstable nucleus of mass 1.7 ✕ 10−26 kg, initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of m1 = 1.0 ✕ 10−27 kg, moves in the positive y-direction with speed v1 = 5.8 ✕ 106 m/s. Another particle, of mass m2 = 9.0 ✕ 10−27 kg, moves in the positive x-direction with speed v2 = 3.8 ✕ 106 m/s. Find the magnitude and direction of the velocity of the third particle. (Assume that the +x-axis is to the right and the +y-axis is up along the page.) magnitude m/s direction ° counterclockwise from the +x-axis