## Mechanics 1

• Kinematics
Motion with constant acceleration, motion in a plane, tangential and radial acceleration, projectile motion, circular motion
• The Laws of Motion
The concept of force, inertial frames, Newton's 2nd and 3rd law, mass and weight, free-body diagrams, uniform and non-uniform circular motion, friction, the fundamental forces
• Work and Energy
Work done by varying forces, kinetic energy, conservative and non-conservative forces, potential energy, conservation of energy.
• Momentum
Conservation of momentum, impulse, elastic and inelastic collisions, the center of mass, rocket propulsion.
• Rotational Motion
Rotational kinematics, torque, angular momentum, rigid-body motion, Kepler's laws
• Oscillatory Motion
Simple harmonic motion, a mass on a spring, a physical pendulum, forced and damped oscillations.

### Problem 1:

A person standing on the surface of Earth throws a ball.  The ball leaves the throwers hand with initial velocity vi and has final velocity vf just before it is caught. If air resistance is negligible, which of the following diagrams correctly represents a possible sequence of velocity vectors for the ball?

### Problem 2:

A spaceship of mass m is initially in a circular orbit around an isolated star of mass M.  The radius of the orbit is R.  If the kinetic energy of the spaceship is doubled by a sort burst of its engine, what is the final total energy of the spaceship?  (The potential energy is taken to be zero when the spaceship is at infinite distance from the star.  G is the gravitational constant.)

(A)  -½GMm/R   (B)  -¼GMm/R   (C)  0   (D)  ¼GMm/R   (E)  ½GMm/R

### Problem 3:

A uniform steel chain is 10 m long with a mass of 2 kg/m.  One end of the chain is attached to a horizontal axis having a radius that is small compared to the length of the chain.  if the chain initially hangs vertically, the work required to slowly wind it up on the axle is close to

(A)  100 J   (B)  200 J   (C)  1000 J   (D)  2000 J   (E)  10000 J

### Problem 4:

The figure shows a small mass connected to a string which is attached to a vertical post.  If the mass is released when the string is horizontal, as shown, the magnitude of the total acceleration as a function of the angle θ is

(A)  g sinθ   (B)  2g cosθ   (C)  2g sinθ   (D)  g(3 cos2θ + 1)½   (E)  g(3 sin2θ + 1)½

### Problem 5:

A car travels with constant speed on a circular road on level ground  In the diagram.  Fair is the force of air resistance on the car.  Which of the other forces best represents the horizontal force of the road on the car's tires?

(A)  FA   (B)  FB   (C)  FC   (D)  FD   (E)  FE

### Problem 6:

Particle 1 of mass m1 and particle 2 of mass m2 = m1/2 are coupled by a massless spring of force constant k and lie at rest on a horizontal frictionless surface.  A third particle of mass m3 = m2 = m1/2 and speed v0 strikes particle 2 and sticks to particle 2.  The peed of the center of mass of the system after the collision is

(A)  v0/4   (B)  v0/3   (C)  v0/2   (D)  v0   (E)  v0 + (8k/(3m1))½

### Problem 7:

In two experiments, two cylinders X and Y are released fro rest at the top of the same inclined ramp and roll down without slipping.  Let tX and tY be the respective times taken for the cylinders to reach a particular line ll' on the ramp.

In the first experiment both cylinders are solid, uniform and of identical dimensions, but they are made of different materials, so that the mass of X is twice that of Y.  MX = 2 MY.  Which of the following relationships is correct?

(A)  tX ≥ 2tY   (B)  2tY > tX > tY   (C)  tX = tY    (D)  2tX > tY > tX   (E)  tX ≥ 2tY

### Problem 8:

In the second experiment both cylinders are solid, uniform and of identical length and density, but the radius of X is twice that of Y.  MX = 2 MY.  Which of the following relationships is correct?

(A)  tX ≥ 2tY   (B)  2tY > tX > tY   (C)  tX = tY    (D)  2tX > tY > tX   (E)  tX ≥ 2tY

### Problem 9:

The period of a simple pendulum is 2π(I/(mgd))½, where I is the moment of inertial about the pivot point and d is the distance from the pivot point to the center of mass.  A circular loop hangs from a nail on a barn wall.  The mass is the hoop is 3 kg and its radius is 20 cm.  If it is displaced slightly by a passing breeze, what is the period of the resulting oscillations?

(A)  0.63 s   (B)  1.0 s   (C)  1.3 s   (D)  1.8 s   (E)  2.1 s

### Problem 10:

An object of mass m is connected to two fixed surfaces by 4 identical springs as shown.  The springs are of negligible mass and have spring constant k.  The period of vertical oscillations is

(A)  2π (2m/k)½     (B)  2π (m/k)½    (C)  2π (2m/(5k))½     (D)  π (m/k)½       (E)  π (m/(2k))½