Plan for Today (AP Physics 2)

Plan for Today (AP Physics 2) • Turn in lab from yesterday • C Test Takers: • Go over MC • Notes/Lecture Variable Forces • Drag Equation Derivation • B Test Takers: • MOPing on computers (pick a problem area)

Calculating Work a Different Way • Work is a scalar resulting from the multiplication of two vectors. • We say work is the “dot product” of force and displacement. • W = F • r • dot product representation • W= F r cos q • useful if given magnitudes and directions of vectors • W = Fxrx + Fyry + Fzrz • useful if given unit vectors

The “scalar product” of two vectors is called the “dot product” • The “dot product” is one way to multiply two vectors. (The other way is called the “cross product”.) • Applications of the dot product • Work W = F  d • Power P = F  v • Magnetic Flux ΦB = B  A • The quantities shown above are biggest when the vectors are completely aligned and there is a zero angle between them.

Why is work a dot product? F  s W = F • r W = F r cos  Only the component of force aligned with displacement does work.

Work and Variable Forces • For constant forces • W = F • r • For variable forces, you can’t move far until the force changes. The force is only constant over an infinitesimal displacement. • dW = F • dr • To calculate work for a larger displacement, you have to take an integral • W =  dW = F • dr

Variable Forces • If force can vary, what should our new equation for work look like? • W = • This would be by a position dependent force

F(x) xb W =  F(x) dx xa x xa xb Work and variable force The area under the curve of a graph of force vs displacement gives the work done by the force.

What if force varies with time? • F = ma • a = dv/dt • a = • F = m = mv • = dx • = dx =

Let’s Integrate that • = dx = • = • = • = • Look familiar? • It’s the Work-Energy theorem • Work is equal to the change in kinetic energy • And it holds constant whether a force is constant or not

What if it’s potential energy • Force of a spring • = - kx • Hooke’s Law • = = • = • ½ k

Spring Potential Energy, Us • Springs obey Hooke’s Law. • Fs(x) = -kx • Fs is restoring force exerted BY the spring. • Ws =  Fs(x)dx = -k  xdx • Ws is the work done BY the spring. • Us = ½ k x2 • Unlike gravitational potential energy, we know where the zero potential energy point is for a spring.

Conservative Forces and Potential Energy • = = Change in potential energy • + Ui • dU = - dx • = - dU/dx

Force and Potential Energy • In order to discuss the relationships between potential energy and force, we need to review a couple of relationships. • Wc = FDx (if force is constant) • Wc =  Fdx = - dU = -DU (if force varies) •  Fdx = - dU • Fdx = -dU • F = -dU/dx

Power • P = dE/dt • Average Power = W/t • P = dW/dt = F * dr/dt = F * v

Forces Reminders • Be sure to draw freebody diagrams • Think about net force and what is going on there

Drag and Resistive forces • Drag is a resistive force proportional to the object’s velocity • How can we express this? • = -bv • v is velocity • b is a constant • Depends on the properties of the medium, shape of the object, size of the object

Considering Drag with other forces • Think about the coffee filter lab. • What forces were acting on the coffee filter? • = mg – bv = ma

Eventually, the coffee filters reached terminal velocity

How can we figure out the terminal velocity? • mg - b = 0 • =

How can we figure out the velocity at any given point? • mg – bv = ma • a = • = g -

Assignment • Solve the equation to get an expression for v for all times t (no ds in there) • = g -

Hint 1: dW = F(r)•dr • Hint 2: ΔU = -Wc (and gravity is conservative!) • Hint 3: Ug is zero at infinite separation of the masses. • Sample problem: Gravitational potential energy for a body a large distance r from the center of the earth is defined as shown below. Derive this equation from the Universal Law of Gravity.

Problem: The potential energy of a two-particle system separated by a distance r is given by U(r) = A/r, where A is a constant. Find the radial force F that each particle exerts on the other.

Problem: A potential energy function for a two-dimensional force is of the form U = 3x3y – 7x. Find the force acting at a point (x,y).

Plan for Today (AP Physics 2)