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Work out problems: 20 points each

1 A simple pendulum (Length L, mass m) is suspended from the ceiling of an elevator. The elevator is accelerating upwards with acceleration a.

1. Draw a Free Body Diagram for the pendulum (5 points)
2. Calculate the component of total force acting on a mass m that makes the pendulum swing. (5 points)
3. find k - assuming that the angle Q is small (5 points)
4. derive the formula for the T - period of oscillation (5 points)

2. A mass m=0.5kg on the end of a spring with the spring constant k=100N/m is set into oscillation by giving it an initial velocity while it is at its equilibrium position. The initial velocity v_o is 0.5m/s

1. find the amplitude of oscillations for initial velocity v₀ (5 points)
2. find the period of oscillations (5 points)
3. what is the acceleration of the mass m when the mass is 0.1 m from the equilibrium position? (5 points)
4. find the amplitude of oscillations if the initial velocity is increased to 2m/s (5 points)

1. Draw Free Body diagram (5 points)
2. Find the minimum acceleration a needed to immerse the wood (7 points)

1. Find the minimum acceleration a needed to immerse the piece of wood if the wood floats in the oil (density 0.8 g/cm³) (8 points)

4. A particle is in simple harmonic motion along the x-axis. The amplitude of the motion is xm. When it is at x = 1/2xm, its kinetic energy is K = 5J and its potential energy (measured with U = 0 at x = 0) is U = 3J. When it is at x = xm, the kinetic and potential energies are:

A) K = 5J and U = 3J B) K = 5J and U = -3J C) K = 8J and U = 0 D) K = 0 and U = 8J E) K = 0 and U = -8J

5. The displacement of a mass oscillating on a spring is given by x(t) = x_mcos(wt +f). If the initial displacement is zero and the initial velocity is in the negative x direction, then the phase constant f is:

A) 0 B)p /2 radians C)p radians D) 3p /2 radians E) 2p radians

6. In simple harmonic motion:

A) The acceleration is greatest at the maximum displacement

B) The velocity is greatest at the maximum displacement

C) The period depends on the amplitude

D) The acceleration is constant

E) The acceleration is greatest at zero displacement

7. Water is streaming downward from a faucet opening with an area of 3.0x10^-5 m2. It leaves the faucet with a speed of 5.0 m/s. the cross sectional area of the stream 0.50 m below the faucet is:

A) 1.5x10^-5 m2 B) 2.0x10^-5 m2 C) 2.5x10^-5 m2 D) 3.0x10^-5 m2 E) 3.5x10^-5 m2

8. Water flows through a constriction in a horizontal pipe. As it enters the constriction, the water's:

A) Speed increases and pressure decreases

B) Speed increases and pressure remains constant

C) Speed increases and pressure increases

D) Speed decreases and pressure increases

E) Speed decreases and pressure decreases

9. A lawn sprinkler is made of a 1.0 cm diameter garden hose with one end closed and 25 holes, each with a diameter of 0.050 cm, cut near the closed end. If water flows at 2.0 m/s in the hose, the speed of the water leaving a hole is:

A) 2.0 m/s B) 32 m/s C) 40 m/s D) 600 m/s E) 800 m/s

10. A fluid is undergoing "incompressible" flow. This means that:

A) The pressure at a given point cannot change with time

B) The velocity at a given point cannot change with time

C) The velocity must be the same everywhere

D) The pressure must be the same everywhere

E) The density cannot change with time or location

11. A U-tube has dissimilar arms, one having twice the diameter of the other. It contains an incompressible fluid and is fitted with a sliding piston in each arm, with each piston in contact with the fluid. When the piston in the narrow arm is pushed down a distance d, the piston in the wide arm rises a distance:

A) d B) 2d C)d/2 D)4d E)d/4

12. The two arms of a U-tube are not identical, one having twice the diameter of the other. A cork in the narrow arm requires a force of 16 N to remove it. The tube is filled with water and the wide arm is fitted with a piston. The minimum force that must be applied to the piston to push the cork out is:

A) 4 N B) 8 N C) 16 N D) 32 N E) 64 N

13. A steel ax and an aluminum piston have the same apparent weight in water. When they are weighed in air:

A) They weigh the same

B) The ax is heavier

C) The piston is heavier

D) Both weigh less than they did in water

E) Depends on their shapes

Ans)       4D 5B  6A   8A   9B   10E   11E   12E    13C