modeling pendulums

A pendulum is modeled by a mass m, attached to one end of a light rod of lenght l, whose other end is attached to an axis. Write down the equation of motion (Newton's Law) modeling the pendulum. Rewrite the model as a system of two first order equations for the position and velocity of the mass. Find the equilibrium points and sketch them in the phase plane. Determine which equilibrium points are stable, which unstable.

Application of Newton's Laws

a.) A wedge with mass M rests on a frictionless horizontal tabletop. A block with mass m is placed on the wedge. There is no friction between the block and the wedge. The system is released from rest. Calculate: i) the acceleration of the wedge ii) the horizontal and vertical components of the acceleration of the block. (Do these answers reduce to the correct results when M is very large?) b) The wedge and block now have a horizontal force P applied to the wedge. What magnitude must P have if the block is to remain at a constant height above the tabletop?

Velocity , Constant Acceleration

1. You and two of your friends decide to have a race. You each start from the center of a circle and travel out along different radius at the same time. One of you moves with a constant speed of v1, the second of you starts from rest and moves with a constant acceleration a, and the third, moving with an initial speed v2, decelerates with constant acceleration -a. Show that if all of you reach the outside of the circle (meaning the distance away from the circle) at the same time then v2 = 2v1. Find the acceleration.

Force and Motion

Several force and motion problems attached.

Ice cooling liquids

DETERMINE HOW MANY BAGS OF ICE ARE REQUIRED TO COOL ONE, 24-BOTTLE CASE OF A BEVERAGE FROM ROOM TEMPERATURE (25ºC) TO A COMFORTABLE, QUAFFING TEMPERATURE (5ºC). ASSUME THE FOLLOWING: THERE IS NO HEAT LOSS FROM YOUR COOLER TO THE AMBIENT ENVIRONMENT ALL HEAT GIVEN OFF BY THE BEVERAGE IS USED TO MELT THE ICE AND/OR HEAT WATER IGNORE THE SPECIFIC HEAT OF THE BEVERAGE CONTAINERS THE INITIAL TEMPERATURE OF THE ICE IS 0ºC THE BEVERAGES ARE IN 12 oz (355 mL) CONTAINERS THE SPECIFIC HEAT OF THE BEVERAGE IS 0.978 cal/g-ºC THE DENSITY OF THE BEVERAGE IS 1.05 g/mL ICE IS SOLD IN 8 POUND BAGS THERE IS NO ICE LEFT OVER AND THE WATER IN THE COOLER IS AT 5ºC WHEN TH... click for more