SO FAR WE HAVE DEALT WITH TWO KINDS OF POTENTIAL ENERGY: GRAVITATIONAL (U=MGH) ELASTIC (U=1/2KX 2 ) POTENTIAL ENERGY GRAPHS CAN PROVIDE INFORMATION ABOUT KINETIC ENERGY, FORCE AND VELOCITY. Potential Energy Curves

Force and Potential Energy We established that the relationship between work and potential energy was: which leads to…

Force and Potential Energy which leads to…. For example: A conservative force always acts to push the system toward a lower potential energy.

A FORCE PARALLEL TO THE X-AXIS ACTS ON A PARTICLE MOVING ALONG THE X- AXIS. THE FORCE PRODUCES A POTENTIAL ENERGY: U(X) = 1.2 X 4. WHAT IS THE FORCE WHEN THE PARTICLE IS AT X=-0.8M? Example

Force and Potential Energy This analysis can be extended to apply to three dimensions:

Potential Energy Curves for a Spring Note: When the spring is either in a state of maximum extension or compression its potential energy is also a maximum When the spring's displacement is DOWN the restoring force is UP When the potential energy function has a negative slope, the restoring force is positive and vice-versa When the restoring force is zero, the potential energy is zero At any point in the cycle, the total energy is constant: U + K = U max = K max

Points of Equilibrium When the force acting on the object is zero, the object is said to be in a state of EQUILIBRIUM! STABLE EQUILIBRIUM – located at minimums, if the object is displaced slightly it will tend back to this location. UNSTABLE EQUILIBRIUM – located at maximums, if the object is displaced slightly it will tend away from this location. STATIC EQUILIBRIUM – located at plateaus, where the net force equals zero.

Points of Equilibrium Stable Equilibrium at x 3 and x 5. Unstable Equilibrium at x 4. Static Equilibrium at x 1 and x 6.

Turning Points Define the boundaries of the particle’s motion. We know that E=K+U, so where U=E, K= 0 J and the particle changes direction. For instance, if E= 4J, there would be turning point at x 2..

If E = 1 J, why is the grey area referred to as an “energy well”? Turning Points

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 What is the value of U(0)? What are the values of x 1 and x 2 ?

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 How much potential energy does the particle have at position x 1 ? If the object was initially released from rest, how fast is it moving as it passes through position x 1 ?

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 How much potential energy does the mass have at x 2 ? How fast is it moving through position x 2 ?

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 Which position, x 1 or x 2, is a position of stable equilibrium?

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 How fast is the particle moving when its potential energy, U(x) = 0 J? If x 3 = ½x 1, then how fast is the particle moving as it passes through position x 3 ?

Example A particle of mass 0.5 kg obeys the potential energy function: U(x) = 2(x - 1) - (x - 2) 3 Sketch the graph of the particle's acceleration as a function of x. Indicate positions x 1 and x 2 on your graph.