AP Physics C Chapter 7

Properties of Conservative Forces

1. The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle. 2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one for which the beginning point and the endpoint and identical).

Vectors A and B expressed as Unit Vectors

A = Ai + Aj + Ak B = Bi + Bj + Bk Scalar Product applied to A and B: A x B = A(x)B(x) + A(y)B(y) + A(z)B(z)

Nonconservative Force

A force is nonconservative if it does not satisfy properties 1 and 2 above. The work done by a nonconservative force is path-dependent. The sum of the kinetic and potential energies of a system in the mechanical energy of the system: E(mech) = K + U

Scalar Product

A x B = ABcos(theta) The scalar product of any two vectors A and B is defined as a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the angle (theta) between them.

Work done by a spring (Spring Force)

F(s) = -kx Where x is the position of a block relative to its equilibrium (x=0), and k is a positive constant force called the force constant or the spring constant.

Relation of force between members of a system to the potential energy of the system

F(x) = - dU / dx The x component of a conservative force acting on a member within a system equals the negative derivative of the potential energy of the system with respect to x. Corresponds to the restoring force in the Spring (Hooke's Law).

Information about vectors A and B

If A is perpendicular to B (theta=90), then A x B = 0. If vector A is parallel to B and the two point in the same direction (theta=0), then A x B = AB. If vector A is parallel to B and the two point in opposite directions (theta=180) then A x B = -AB. The scalar product is negative 90 <= theta <= 180.

Work is an energy transfer

If W is the work on a system and W is positive, energy is transferred to the system; if W is negative, energy is transferred from the system.

Work done by a Spring

If the block undergoes a displacement from x = x(i) to x = x(f), the work done by the spring force on the block is: W(i) = ∫ (-kx) * d(x) = 1/2kx(i)^2 - 1/2kx(f)^2

Kinetic Energy

K = 1/2mv^2 Kinetic Energy is scalar and has the same units as work (J).

Stable, Unstable, and Neutral Equilibrium

Systems can be in three types of equilibrium configurations when the net force on a member of the system is zero. Configurations of stable equilibrium correspond to those for which U(x) is a minimum. Configurations of unstable equilibrium correspond to those for which U(x) is a maximum. Neutral equilibrium arises when U is constant as a member of the system moves over some region. :]

Hooke's Law

The force required to stretch or compress a spring is proportional to the amount of stretch or compression x.

Gravitational Potential Energy

U(g) = mgy Scalar and has the same units as work (J). Only valid for objects near the Earths surface where g is approximately constant.

Elastic Potential Energy

U(s) = 1/2kx^2 Energy stored in a deformed spring (one that is either compressed of stretched from its equilibrium position).

Distributive law of multiplication

The scalar product obeys the Distributive law of multiplication: A x (B + C) = A x B + A x C

Sign of Work

The sign of work depends on the direction of F relative to (theta)r. The work done by an applied force is positive when F onto (theta)r is in the same direction as the displacement. When the F onto (theta)r is in the opposite direction of the displacement, work is negative.

Work

W = F x (delta)r x cos(theta) Work done on a system by a constant force is the product of the magnitude of the Force, the magnitude of the Displacement (delta r), and Cos(theta), where theta is the angle between the force and displacement vectors.

Work expressed as a scalar product

W = F x (delta)r x cos(theta) = F x (delta)r

Work done on a particle

W = F x (delta)x

Work-Kinetic Energy Theorem

When work is done on a system and the only change in the system is in its speed, the net work done on the system equals the change in kinetic energy of the system, and is expressed as: W = (delta)K. If speed increases then the work is positive; if speed decreases then the work is negative.

More information about Work

Work is scalar. The displacement is that of the point of application of the force. The work done by the normal force, and the work done by the gravitational force are both zero because the forces are perpendicular to the displacement and have zero components along an axis in the direction of (delata)r.

Scalar products of Unit Vectors

i x i = j x j = k x k = 1 i x j = i x k = j x k = 0

SI unit of Work

newton x meter. (N x m = kg x m^2 / s^2) Called joules (J).

Work done for a graph with intervals

x(f) W = Σ F(x) * (delta)x x(i)

Work done by F(x) on the system of the particle as it moves from x(i) to x(f)

x(f) W = ∫ F(x) * d(x) x(i)