Work and Energy A calculus-based perspective AP Physics C

The old “special case” W=F d is a special case W=F d is a special case F must be in the direction of motion F must be in the direction of motion F is a constant force F is a constant force What if F is not parallel to d? What if F is not parallel to d? The Dot or Scalar Product. The Dot or Scalar Product. One vector times another to “make” a scalar One vector times another to “make” a scalar W= Fcos θ d W= Fcos θ d

If the graph below shows the force exerted by the Death Star’s Tractor beam. How could you calculate the work done on a ship being pulled from position a to b? A. Find the slope of the line between a and b B. Find the Area under the curve from point a to b C. Multiply force x (b-a)

Area Under the Curve For Hooke’s Law, the force is linear. For Hooke’s Law, the force is linear. For a force vs. distance graph, the work is just the area under the curve. The shape is a triangle. For a force vs. distance graph, the work is just the area under the curve. The shape is a triangle. What if the area under the curve is not geometric? What if the area under the curve is not geometric?

Finding an integral is a way of accurately finding the area under a curve. My looking at smaller and smaller “pieces” of x, multiplying them by the force, and summing them we can find total area. Most Forces aren’t constant. Springs Magnetic fields Pushes/pulls Gravity over large distances The Reality

Integration Rules!!

Example: Work done to stretch a spring. F(x) = kx F(x) = kx Hooke’s Law Hooke’s Law

How much work do you do if you stretch a spring from an initial position of 0m to the 1m position if the spring constant is 49 N/m A. 49 Nm B. 25 Nm C. 98 Nm D. 10 Nm

Work for a varying force

A force F(x) acts on a particle. The force is related to the position of the particle by the formula F(x) = Cx 3, where C is a constant. Find the work done by this force on the particle when the particle moves from x = 1.5 m to x = 3 m. A. 1 J B. 34 J C. 0 J D. 19C J

Solution

Energy

Work-Energy Theorem The change in the kinetic energy of an object is equal to the net work done on the object. The change in the kinetic energy of an object is equal to the net work done on the object.

Types of Force Conservative Conservative Obeys conservation of energy Obeys conservation of energy Examples Examples Spring force Spring force Gravity Gravity Non-conservative Energy is transferred into non-mechanical forms Examples Friction Air drag

Equilibrium Occurs when net force = 0 Occurs when net force = 0 If force = F(x), then equilibrium exists at points where F(x) = 0. If force = F(x), then equilibrium exists at points where F(x) = 0. Equilibrium exists where dU/dx = 0 Equilibrium exists where dU/dx = 0

Power

Power takes many forms