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Inclined Planes Block on a Ramp, Example • Axes are rotated as • Choose the usual on an incline coordinate • The direction of system with x impending motion along the incline would be down the and y plane perpendicular to • acts up the the incline plane • Replace the – Opposes the motion of with its • Apply Newton’s components Laws and solve equations

Friction/Inclined Plane Example Multiple Objects – Example

• When you have more than one object, the problem-solving strategy is applied to each A 4000 kg truck is parked on a 15º slope. object How big is the frictional force on the • Draw free body diagrams for each object truck? • Apply Newton’s Laws to each object • Solve the equations

Multiple Objects – Example, cont. A 5-kg block hangs vertically by a massless string from a . The string is connected over the pulley to a 7-kg block on a ramp inclined at a particular angle. If the angle of the ramp is 28º and there is no friction, what is the acceleration of the blocks?

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Springs & Hooke’s Law The spring constant

• Springs can supply (similar to • k tells you something about the physical strings & tension) properties of the spring • These are restoring forces: the spring • Big value of k: spring is difficult to stretch/ wants to be in its equilibrium position compress • Force due to a spring depends on how • Small value of k: spring is easy to stretch/ “strong” the spring is: spring constant k compress, very flexible • Also depends on how much the spring is stretched/compressed: x

Fspring = -kx

Uniform Circular Motion Centripetal Acceleration

• When moving at a constant speed in a • An object traveling in a circle, even though circular path, acceleration is non-zero it moves with a constant speed, will have because direction of velocity changes an acceleration • The centripetal acceleration is due to the 2 ac = v /r change in the direction of the velocity

2 ac = centripetal acceleration (m/s ) v = speed of object (m/s) r = radius of circle (m)

Examples of centripetal Level Curves acceleration: • Ball-on-a-string • Friction is the • Car on a banked circular road force that • Car on a circular track produces the • Planet orbiting a star centripetal • Car driving over a circular hill acceleration • Car driving through circular valley • Roller coaster loop

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Example: A 1500kg car takes a Period & Frequency sharp turn with a radius of 20.0 m. If the coefficient of friction between • Period (T): Time to complete one full the tires and the road is 0.80, rotation what’s the fastest speed at which • Frequency (f): Number of rotations the car can take the turn? completed per second. • f = 1/T, T = 1/f • v = 2πR/T

Work and Energy Energy

• : W = F!d (picks out • Several different varieties of energy, depending parallel components) on what the object is doing, where it is located, • F " Force on object what it interacts with, etc. • d " displacement • Kinetic Energy (motion): KE = ½ mv2 • Direction of force relative • Rolling Kinetic Energy: KE = ½ Iω2 to motion matters! rot • Gravitational Potential Energy: Ug = mgh • Elastic Potential Energy (springs): U = ½ kx2 • Units: 1 Joule = 1 kg m2/s2 sp • All have the same units as work (Joules)

Methods of Energy Transfer: Work-KE Theorem

• By work from a force • Work is done on or by objects/systems • By radiating heat • Energy is an internal property that objects/ systems posess • By wave propagation (either physical waves or electromagnetic waves) W = ΔKE 2 2 F ! d = ½ mfvf – ½ mivi • All energy comes from/goes SOMEwhere! It’s never destroyed, but can become less Work (+ or -) done on an object causes “useful”. changes in its KE (increase or decrease).

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Example: Loading a Ship Types of Forces

A 3,000 kg truck is loaded onto a • There are two general kinds of forces ship by that exerts upward – Conservative: Work and energy associated force of 31,000 N on truck. This with the force can be recovered (Example: force is applied over a distance of Gravity 2.0 m. – Nonconservative: forces are dissipative and (a) Find work done on truck by work done against it cannot easily be crane recovered (Example: Friction) (b) Find work done on truck by gravity. (c) Find net work done on the truck.

Conservative Forces More About Conservative Forces

• A force is conservative if the work it does • Examples of conservative forces include: on an object moving between two points is – Gravity independent of the path the objects take – Spring force between the points – Electromagnetic forces – The work depends only upon the initial and • Potential energy is another way of looking final positions of the object at the work done by conservative forces – Any conservative force can have a potential energy function associated with it

Nonconservative Forces Friction Depends on the Path • A force is nonconservative if the work it does on an object depends on the path • The blue path is taken by the object between its final and shorter than the red starting points. path • The work required is • Examples of nonconservative forces less on the blue path – kinetic friction, air drag, propulsive forces than on the red path • Friction depends on the path and so is a non-conservative force

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Work-Energy Theorem, Extended , cont.

• The work-energy theorem can be extended to • Total mechanical energy is the sum of the include potential energy: kinetic and potential energies in the W (KE KE ) (PE PE ) system nc = f − i + f − i E = E • If other conservative forces are present, potential i f energy functions can be developed for them and KEi + PEi = KEf + PEf their change in that potential energy added to the right side of the equation – Other types of potential energy functions can be added to modify this equation

Problem Solving with Conservation Problem Solving, cont of Energy • Define the system • Verify that only conservative forces are • Select the location of zero gravitational potential present energy – Do not change this location while solving the problem • Apply the conservation of energy equation • Identify two points the object of interest moves to the system between – Immediately substitute zero values, then do – One point should be where information is given the algebra before substituting the other – The other point should be where you want to find out values something • Solve for the unknown(s)

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