Potential Energy and Conservation of Energy

8.01 W05D2

Next Reading Assignment: W05D3

Quiz 4: Circular Motion

2

Concept Question: Marble Run

Concept Question: Marble Run Shortest Time

Strategy: Using Multiple Ideas

1. Energy principle:

2. For circular motion, may need a constraint condition based on Newton’s Second Law in the radial direction

Kinitial +Uinitial = K final +U final + Elost due to non−conservative forces

Worked Example: Loop-the-Loop

An object of mass m is released from rest at a height h above the surface of a table. The object slides along the inside of the loop-the-loop track consisting of a ramp and a circular loop of radius R shown in the figure. Assume that the track is frictionless. When the object is at the top of the track (point a) it just loses contact with the track. What height was the object dropped from?

Demo slide: Loop-the-Loop B95 http://scripts.mit.edu/~tsg/www/

index.php?page=demo.php?letnum=B 95&show=0

A ball rolls down an inclined track and around a vertical circle. This demonstration offers opportunity for the discussion of dynamic equilibrium and the minimum speed for safe passage of the top point of the circle.

Demo slide: potential to kinetic energy B97

http://scripts.mit.edu/~tsg/www/index.php?page=demo.php?letnum=B 97&show=0

This demonstration consists of dropping a ball and a pendulum released from the same height. Both balls are identical. The vertical velocity of the ball is shown to be equal to the horizontal velocity of the pendulum when they both pass through the same height.

Table Problem: Spring Gun A small block of mass m is pushed against a spring

with spring constant k and held in place with a catch. The spring compresses an unknown distance x. When the catch is removed, the block leaves the spring and slides along a frictionless circular loop of radius r. When the block reaches the top of the loop, the force of the loop on the block (the normal force) is equal to twice the gravitational force on the mass. How far was the spring compressed?

Table Problem: Extreme Skier An extreme skier is accelerated from rest by a spring-action cannon, skis

once around the inside of a vertically oriented circular loop, then comes to a stop on a carpeted up-facing slope. Assume the cannon has a spring constant k and a cocked displacement x0 , the loop has a radius R , and the slope makes an angle θ to the horizontal. The only surface with friction is the carpet, represented by a friction constant µ. Gravity acts downward, with acceleration g, as shown. What is the linear distance d the skier travels on the carpet before coming to rest?

Worked Example: Block Sliding off Hemisphere

A small point like object of mass m rests on top of a sphere of radius R. The object is released from the top of the sphere with a negligible speed and it slowly starts to slide. Find an expression for the angle θ with respect to the vertical at which the object just loses contact with the sphere.

Table Problem: Energy Diagram

The figure above shows a graph of potential energy U(x) verses position for a particle executing one dimensional motion along the x-axis. The total mechanical energy of the system is indicated by the dashed line. At t =0 the particle is somewhere between points A and G. For later times, answer the following questions.

a)  At which point will the magnitude of the force be a maximum? b)  At which point will the kinetic energy be a maximum? c)  At how many of the labeled points will the velocity be zero? d)  At how many of the labeled points will the force be zero?

Table Problem: Potential Energy Diagram

A body of mass m is moving along the x-axis. Its potential energy is given by the function U(x) = b(x2-a2) 2 where b = 2 J/m4 and a = 1 m .

a) On the graph directly underneath a graph of U vs. x, sketch the force F vs. x.

b) What is an analytic expression for F(x)?