Last Time: Work, Kinetic Energy, Work-Energy Theorem Today: Gravitational Potential Energy, Conservation of Energy, Spring Potential Energy, Power HW #4 extension to Friday, 5:00 p.m. HW #5 now available Due Thursday, October 7, 11:59 p.m.

Conceptual Question A B C D A 10 N force acts on a block, as shown below (other forces may also be acting). The block moves the same horizontal distance D in the +x-direction in all four cases below. Rank the amount of work done by the 10 N force, in order of most positive, to most negative. A B C D θ y x

Conservative vs. Non-Conservative Forces In general, there are two kinds of forces : “Conservative” Forces “Non-Conservative” Forces Energy can be recovered Energy cannot be recovered E.g., Gravity E.g., Friction Generally: Dissipative

Gravitational Potential Energy Suppose an object falls from some height to a lower height. How much work has been done by gravity ? y |F| |Δy| cos θ Δy mg yi If an object is raised to some height, there is the “potential” for gravity to do positive work. Positive work means an increase in the object’s kinetic energy. yf

Gravitational Potential Energy So we then define the “gravitational potential energy” y: vertical position relative to Earth’s surface (or another reference point) Gravitational Potential Energy PE = mgy SI unit: Joule The gravitational potential energy quantifies the magnitude of work that can be done by gravity. By the Work-Energy Theorem, the gravitational potential energy is then equal to the change in the object’s kinetic energy if it falls a distance y.

Reference Level for Potential Energy We have defined the gravitational potential energy to be: Q: Does it matter where we define y = 0 to be ? A: No, it doesn’t matter. All that matters is the difference in the potential energy, ΔPE = mg Δy . It doesn’t matter where we define zero to be. 100 m 5 m In both of these, the object falls 5 m. 95 m 0 m

Gravity and Conservation of Energy Conservation Law : If a physical quantity is “conserved”, the numerical value of the physical quantity remains unchanged. Conservation of Mechanical Energy : Sum of kinetic energy and gravitational potential energy remains constant at all times. It is a conserved quantity. If we denote the total mechanical energy as E = KE + PE, the total mechanical energy E is conserved at all times.

Gravity and Conservation of Energy Ignoring dissipative forces (air resistance), at all times the total mechanical energy will be conserved : initial total mechanical energy final total mechanical energy

Example A 25-kg object is dropped from a height of 15.0 m above the ground. Assuming air resistance is negligible … (a) What is its speed 7.0 m above the ground ? (b) What is its speed when it hits the ground ?

Example A skier starts from rest at the top of a frictionless ramp of height 20.0-m. At the bottom of the ramp, the skier encounters a horizontal surface where the coefficient of kinetic friction is μk = 0.210. Neglect air resistance. Find the skier’s speed at the bottom of the ramp. (b) How far does the skier travel on the horizontal surface before coming to rest ?

Pendulum and Conservation of Energy B A pendulum is released from rest at point A. Ignoring friction … What is its speed at the bottom of its trajectory at B ? How high does it swing on its way up to C ? When it swings back to A, does it return to its initial height ?

Springs One must do work on a spring to compress or stretch it. The work it takes to compress or stretch the spring can be recovered as kinetic energy. This means we can find a potential energy function for springs, which we can then use in the Work-Energy Theorem.

Hooke’s Law x = 0 : Position of spring when not compressed/stretched Force exerted by spring : compressed F = –kx k: “spring constant” If compressed, x < 0, so F > 0 ! x

Hooke’s Law x = 0 : Position of spring when not compressed/stretched Force exerted by spring : stretched F = –kx x If stretched, x > 0, so F < 0 !

Spring Potential Energy x = 0 x = 0 : compressed stretched x x If spring is compressed or stretched, it will exert a force, and so it has the potential to do work. Elastic potential energy associated with this spring force is : k : “spring constant” x : displacement of spring SI Unit: Joules

Springs and Conservation of Energy Assuming only conservative forces (i.e., no non-conservative forces, such as friction), systems with springs will obey : Initial KE Initial Spring PE Initial Grav. PE Final KE Final Spring PE Final Grav. PE

Non-Conservative Forces If there is a non-conservative force (e.g., friction) acting, work done by this force is : Mechanical energy changes. Work done by non-conservative force dissipated as, e.g., heat.

Example 5.9 (p. 137) A block with mass of 5.0 kg is attached to a horizontal spring with spring constant k = 400 N/m. The surface the block rests on is frictionless. If the block is pulled out to xi = 0.05 m and released … (a) Find the speed of the block when it reaches the equilibrium point (x = 0). (b) Find the speed when x = 0.025 m. (c) Repeat (a) if friction acts, with μk = 0.150.

Power 20 Watt CFL Light Bulb 300 hp Ford Mustang If an external force does work W on an object in some interval Δt, then the average power delivered to the object is the work done divided by the time interval W in Joules Δt in seconds P in Watts = Joule/second The higher the power, the more work that can be done in a given time interval.

Power Note that we can write : Average power is a constant force times the average speed. Note on units : 1 Watt = 1 Joule/second = 1 kg-m2/s3

Example: 5.53 The electric motor of a model train accelerates the train from rest of 0.620 m/s in 0.021 s. The total mass of the train is 0.875 kg. Find the average power delivered to the train during its acceleration.

Next Class 6.1 – 6.2 : Momentum, Impulse, Conservation of Momentum