Work & Energy

a constant force that moves an object a distance s in the same direction as the force does

work

• W = Fs, and the work of a constant force is parallel to the direction of motion

• the work done by the applied force is larger if either the magnitude of the force is increased or the distance that the force acts over is larger

work is a

scalar quantity (no direction) and independent of the direction of motion (positive quantity)

• the SI units of work are Joules, which are N x m = kg m²/s²

  • one J is the work of a force of one N applied for one m

if the constant force acts in a direction other than that of displacement,

only the force component in the direction of displacement will do work

• W = (Fcosθ)s

the component of the force perpendicular to the displacement (Fsinθ) is

helping the floor support the weight of the object, not helping to move the object across the floor

for a free body diagram of an object with four forces applied on the ground

• Wn = (Ncos90)s = 0 (normal force does 0 work)

• Wg = (Fgcos90)s = 0 (gravitational force does 0 work)

• Wf = (Pcos0)s = Ps (positive parallel work for motion)

• Wfk = (fkcos180)s = -fks (negative work done by friction, which hinders the motion of the object in the opposite direction of its motion)

kinetic energy

an object of mass m moving with a speed v has a kinetic energy K defined as the scalar quantity K = ½ mv²

• the kinetic energy of an object is always positive, since the speed and mass of the object are positive scalar quantities

• the SI units for kinetic energy are Joules = N x m = kg m²/s²

kinetic energy measures the combined effect of

having a mass and moving at a certain speed

• kinetic energy is proportional to the mass of the object (more kinetic energy occurs from an object that has a greater mass if it is moving at the same speed)

  • an object with a certain mass has the same kinetic energy regardless of direction, as long as it moves at the same speed

work energy theorem

the work done by the net force on an object is equal to the change in the kinetic energy of the object

• W = ∆K = Kf - Ki = ½ mvf² = ½ mvi² = ½ m (vf² - vi²)

if the net force Fnet = 0, the work done is also

W = 0, but the kinetic energy would not change (mechanical equilibrium)

• if W > 0, ∆K > 0

• if W < 0, ∆K < 0

• the sign of the work of the net force is equal to the sign in the change of the kinetic energy of the object

  • if W is positive, the object is speeding up

  • if W is negative, the object is slowing down

net work done by force and kinetic friction on a flat surface

• Wf = Facos0 s = Fa∆x

• Wfk = fkcos180 s = -fk ∆x

  • Wnet = Wf + Wfk = Fa∆x - fk ∆x = ∆x (Fa - fk)

net work done by force and kinetic friction on an inclined surface

• Wf = Facosθ ∆x

• Wfk = fkcos180 ∆x = -fk ∆x

• Wg = mgsinθ ∆x

  • Wnet = Wf + Wg + Wfk = Facosθ ∆x + mgsinθ ∆x - fk∆x = ∆x (Facosθ + mgsinθ - fk)

if an object is let go, gravity has the

ability or potential to move the object all the way to the ground

• if the object has dropped to a new height hf from h0, the work done by gravity Fg is Wg = Fgs = mg (h0 - hf) > 0

• once the object is at the ground, gravity cannot do any more work

  • there is no more displacement of the object possible, thus there is no more potential for gravitational work

gravitational potential energy (U) is a

kind of energy that an object has by virtue of being at a certain height above Earth’s surface

• U is equal to the work an external agent would need to do against the force of weight to put the object at its height, h

  • U = mgh

  • function of height or displacement and U is proportional to the height and mass of the object

  • U is positive at every point above the floor

the object’s position is measured from an

arbitrary height, at which gravitational potential energy is set to 0

• it is convenient to take the lowest point in the range of motion, so U > 0 everywhere else

a change in gravitational potential energy is given by

∆U = Uf - Ui, and is only affected by changes in the height difference between initial and final location

• it does not depend on the specific path taken

• left or right displacements are irrelevant in terms of the work done