Classical Mechanics Review 4: Units 1-22 Midterm 4, Friday December 2 Units 1-22 Two 4x6 cards

Important Equations t = rF sinq Gravitational Potential Energy of a Rigid Body Statics: and about ANY axis Dynamics: and Kinetic Energy of a Rigid Body rotating about a fixed axis t = rF sinq Angular Momentum of a Particle: Angular Momentum of a Rigid Body Simple Harmonic Oscillations Spring: Physical Pendulum:

Example: Pole Supported by a Wire A pole of mass M and length L is attached to a wall by a pivot at one end. The pole is held at an angle θ above the horizontal by a horizontal wire attached to the pole at its other end. The moment of inertia of the pole is ICM = ML2/12. (a) What is the tension in the wire? (b) What are the vertical and horizontal components of the force R on the pole at the pivot? (c) Now the wire breaks. What is the initial angular acceleration α of the pole? (d) Find the angular speed of the pole just before it hits the wall. θ wire pole wall

Example: Pole Supported by a Wire A pole of mass M and length L is attached to a wall by a pivot at one end. The pole is held at an angle θ above the horizontal by a horizontal wire attached to the pole at its other end. The moment of inertia of the pole is ICM = ML2/12. θ wire pole wall

Example: Person on a Beam A uniform horizontal beam with a length l and mass M is attached to a wall by a pin connection. Its far end is supported by a cable that makes an angle ϕ with the beam. If a person of mass m stands at a distance d from the wall, find the tension in the cable T and the force H exerted by the wall on the beam.

Example: Rotating Rod A rod of length L and mass M is attached to a frictionless pivot and is free to rotate in the vertical plane. The rod is released from rest in the horizontal position. ICM = ML2/12 (a) What is the moment of inertia of the rod about its left end? (b) What is the angular speed when it reaches its lowest position (c) What is the force R the pivot exerts on the rod at that point? IP = ICM + M(L/2)2 = ML2/3

Example: Falling Rod (a) The rotational kinetic energy of the rod. A rod of length L and mass M is pivoted about a horizontal, frictionless pin through one end. The rod is released, almost from rest in a vertical position. The moment of inertia of the rod about its center is Icm = ML2/12. At the instant the rod makes an angle of θ with the vertical find: (a) The rotational kinetic energy of the rod. (b) The angular acceleration of the rod. (c) The speed of the center of mass of the rod. L m M θ P

Example: Leaning Beam A uniform beam of length L and mass M is leaning against a frictionless vertical wall. The bottom of the beam makes an angle θ with the horizontal ground. ICM = ML2/12. (a) Assuming the beam is in static equilibrium, what is the magnitude of the frictional force F between the beam and the ground? (b) What is the minimum coefficient of static friction required so that the beam does not slip? θ F

Example: Falling disk A solid uniform disk of radius R and mass M is initially supported by a force from the pivot and another vertical force F at the other end of the disk as shown. (ICM = MR2/2) 1. Find the force F and the force the pivot exerts on the disk. 2. Now the force F is removed and the disk is released from rest. It is free to rotate about the pivot in the presence of gravity. Find the angular speed of the disk at the lowest point. 3. Find the speed of the center of the disk at the lowest point. 4. Find the components of the force R at the pivot at that point. F

Example: Falling disk A solid uniform disk of radius R and mass M is initially supported by a force from the pivot and another vertical force F at the other end of the disk as shown. (ICM = MR2/2) F

Example: Collision with a Disk A bullet of mass m = 0.2 kg is flying at the constant velocity v = 10 m/s in x direction and at a distance of d = 0.4 m from the x-axis towards a vertical solid disk of mass M = 1kg that is rotating counterclockwise about the fixed axis with the angular velocity ω0 = 10 rad/s. The bullet strikes the disk and sticks at point A. The radius of the disk is 0.5 m. For the disk: Icm = MR2/2. (a) What is the angular momentum of the bullet about the center of the disk? (b) What is the angular velocity ω of the disk right after the collision? (c) What would the angular velocity of the disk be, when the bullet has rotated to point B after the collision?   V=10m/s x m=0.2kg M=1 kg d = 0.4 m A w0 = 10 rad/s R=0.5 m B After the collision energy is conserved

Another Playground Example A horizontal circular platform of mass M and radius R rotates freely about a vertical axle. A student of mass m walks slowly from the rim of the disk towards its center. The moment of inertia of the platform is Idisk = MR2/2. If the angular speed of the system is ωi when the student is at the rim, what is the angular speed when she reaches a point r from the center.

Example: Rod and Disk A solid disk of mass m1 and radius R is rotating with angular velocity ω0. A thin rectangular rod with mass m2 and length l = 2R begins at rest above the disk and dropped on the disk where it begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2. (a) What is the final angular momentum of the rod-disk system? (b) What is the final angular velocity of the disk? (c) What is the final kinetic energy of the system? m2 m1 w0 wf

Example: Rod and Disk A solid disk of mass m1 and radius R is rotating with angular velocity ω0. A thin rectangular rod with mass m2 and length l = 2R begins at rest above the disk and dropped on the disk where it begins to spin with the disk. ICM,rod = Ml2/12, ICM,disk = MR2/2. The rode takes a time Δt to accelerate to its final angular speed. What average torque is exerted on the rod? m2 wf

Example: Block-Spring system A block of mass m connected to a light spring with spring constant k oscillates on a horizontal frictionless air track. (a) Calculate the total energy of the system and the maximum speed of the block vmax if the amplitude is A. (b) What is the velocity v of the block when its position is x?

Example: Simple Pendulum A simple pendulum with mass m and length L hangs from the ceiling. It is pulled back to an small angle of θmax from the vertical and released at t = 0. a) What is the magnitude of the force on the pendulum bob perpendicular to the string at t = 0? b) What is the angular displacement as a function of time? c) What is the maximum speed of the mass m? d) What are tangential and radial accelerations of m as the pendulum passes through the equilibrium position? e) What would be the period of oscillation if the mass m is a sphere of radius R. The moment of inertia of a sphere about its center of mass is Icm= 2mR2/5. q L

Example: Simple Pendulum A simple pendulum with mass m and length L hangs from the ceiling. It is pulled back to an small angle of θmax from the vertical and released at t = 0. q L

Example: Oscillating Hoop A pendulum is made by hanging a thin hula-hoop of radius R and mass M on a small nail. a) What is the period of oscillation of the hoop for small displacements? b) Use energy methods to find the maximum angular speed of the hoop if its is displaced an angle θmax from the vertical position and released from rest? pivot (nail) D

Example: Block-Spring system A block of mass m connected to a light spring with spring constant k is free to oscillate on a horizontal frictionless surface. The block is displaced xi from equilibrium to the right and given an initial speed vi. (a) What are the period, amplitude and phase constant of its motion? (b) What is the maximum speed and acceleration of the block? (c) Express the position, velocity and acceleration as a function of time. m k

Example: Block-Spring system A block of mass m connected to a light spring with spring constant k is free to oscillate on a horizontal frictionless surface. The block is displaced xi from equilibrium to the right and given an initial speed vi. From the initial conditions: xi = Acos(-f) vi = −Aωsin(-f) m k