amplitude of the oscillations = larger amplitude increases max. speed

larger amplitude = larger restoring force

larger restoring force = larger acceleration

stiffness of string = stiffer string increases max. speed

stiff spring with high spring constant = stronger restoring force

again, larger restoring force = larger acceleration

mass of the object = larger mass decreases max. speed

the deformation of an object is proportional to the force causing the deformation.

formula for the force required to deform a spring according to how deformed it is (the displacement) F = kx

ie. force is proportional to displacement; more stress = more strain

in a graph for force vs displacement, the slope (force⁄position aka N⁄m) is equal to k, the spring constant. see picture below for graph example!

spring constant is also known as force constant

larger mass will oscillate more slowly (large period) and a stiff spring will oscillate more quickly (small period)

restoring force always acts in the opposite direction to the displacement

restoring force = -kx

a spring moves fastest when it goes through the equilibrium position.

as we know, when an object is moving at constant speed, W = F∆d and P = W⁄t

if we insert W into the P equation, it becomes P = F∆d⁄t

efficiency = the ratio of the energy output of a system to the energy input of a system

efficiency = useful work / total energy expended

can be measured in terms of power or energy

another way of looking at this:

work done = the change in net energy of an object. measured in joules.

power = the rate that work is being done at; or, the amount of work done in a period of time:

power is measured in watts and horsepower.

watt = W = J/s

horsepower = hp = 746 W

as illustrated in the above picture, power is affected by the speed of an object. the faster an object moves, the more power it consumes. you can prove this mathematically:

conservative forces

(eg. force of gravity, elastic forces)

don’t affect the mechanical energy of a system

path independence (the total work done isn’t affected by the path of the object’s motion)

work = change in potential energy (this is called the work-potential energy theorem)

W = ∆Ep = mg∆h

if the object being worked on also gains kinetic energy, work = change in potential energy + change in kinetic energy (this is the work-energy theorem)

W = ∆Ep + ∆Ek = mg∆h + (1⁄2)mv2

therefore… the car in the diagram below (if we assume it’s frictionless) will have the same speed at point Q no matter which ramp it glides down!

non-conservative forces

(eg. friction)

cause the energy of the system to change; thus energy is not conserved (no-brainer)

causes energy to leave the system via heat or sound

work done by friction = the product of the force of friction and the distance of the path traveled by the object

Wf = Ff∆d

therefore, in the same diagram above, because Ramp B has a longer distance, the car will lose more energy on Ramp B. and since potential energy will always be the same at the bottom (0), this loss in energy affects only the kinetic energy of the car.

the calculated value of Wf should always be negative, because friction always reduces the mechanical energy.

the amount of work done by friction thusly affects the total mechanical energy of the system:

but not all non-conservative forces reduce the mechanical energy within a system!

eg. motors add energy to a system

Em2 = Em1 + the total work done by all the external forces

pendulums = good example of an isolated system.

at its highest point, the pendulum has no Ek and lots of Ep.

at its lowest point, it’s the opposite.

however, mechanical energy at all points throughout its path is the same, whether it is the sum of Ek and Ep or just one or the other.