Instructor: Dr. Tatiana Erukhimova Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 39, 40, 41 Hw: Chapter 15 problems and exercises

For symmetrical objects rotating about their axis of symmetry: Second Law:

The rope is assumed not to slip as the pulley turns The rope is assumed not to slip as the pulley turns. Given m1, m2, R, and I find the acceleration of mass m1. I m1 m2 R

What is the moment of inertia of a disk of thickness h, radius R and total mass M about an axis through its center?

A cylinder rolls down an inclined plane of angle θ without slipping. a) Find the acceleration of the cylinder. b) Find the maximum θ if the coefficient of static friction is μ.

Rotational Kinetic Energy

R O F

Two masses, and , are attached by a massless, unstretchable string which passes over a pulley with radius R and moment of inertia about its axis I. The horizontal surface is frictionless. The rope is assumed NOT to slip as the pulley turns. Find the acceleration of mass .

Gyroscopic precession The precession of a gyroscope shows up in many “common” situations.

A rotating flywheel

Ch. 17 Statics

There is a ladder against a wall There is a ladder against a wall. There is no friction at the wall and the coefficient of friction at the floor is . At what angle can the ladder be placed so that it would slip if the angle were decreased infinitesimally?

The advantage of a lever GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH                                                  

Simple harmonic oscillator

Lots of stuff Vibrates or Oscillates: Examples Lots of stuff Vibrates or Oscillates: Radio Waves Guitar Strings Atoms Clocks, etc…

Why do we care? Lots of engineering problems are oscillation problems Buildings vibrating in the wind Motors vibrating when running Solids vibrating when struck Earthquakes

Resonance

The forces No force Force in –x direction Force in +x direction

More Detail Time

Some Terms Amplitude: Max distance Period: Time it takes to get back to here

A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance L and let go at t=0 Find the position of the mass at all times Where does the maximum speed occur? What is the maximum speed?

If the block is pulled a distance x1 to the right and released from rest, how long will it take the block to return to its equilibrium position? How does this time change if the displacement is increased from x1 to 2x1? How fast will the block be moving at its equilibrium position for the x1 displacement?

A bullet of mass m is fired with velocity of magnitude into a block of mass M. The block is connected to a spring constant k and rests on a frictionless surface. Find the velocity of the block as a function of time. (Assume the bullet comes to rest infinitely quickly in the block, i.e. during the collision the spring doesn’t get compressed.)

Simple pendulum How long does it take to return to the equilibrium?

Have a great day! Hw: Chapter 18 problems and exercises