Work and Energy Chapter 6

Mechanical work and mechanical energy Energy provides a convenient way to relate position and velocity magnitudes for situations where • Force magnitudes and/or directions change • Acceleration magnitude and/or direction changes (You can't use the kinematic equations for constant accelerated motion when the acceleration is not constant.) Examples: Compressing a spring requires an increasing force Swinging pendulum has variable force directions Energy calculations ignore the time it takes for the object to move from initial to the final locations.

displacement s Calculating mechanical work for a constant force that is in the same direction as the displacement. Mechanical work is a scalar quantity (no direction). s is the displacement Units:

Mechanical work for a constant force Work is done by the component of the force that is in the direction of the displacement. is the force component in the direction of the displacement

Example 1 Pulling a suitcase 50° 75 m Example 1 Pulling a suitcase Find the work done by a 45 N force that pulls at a 50° angle if the horizontal displacement is 75 m.

cosine values positive value for work negative value for work Perpendicular forces do zero work.

Total mechanical work The total work done is the work done by all of the forces. This value is the same as the work done by the net force If the displacement s in the same direction as the net force then the angle θ is zero. ( cos 0° = 1 ) Total work done on the object is

velocity Kinetic energy Using a lot of algebra…….. KE is the kinetic energy for an object of mass m and velocity v velocity Kinetic energy equation for v KE is a scalar quantity. Units: J

Work-Energy Theorem When mechanical work is done on an object, the kinetic energy of the object changes according to this rule….

Example 4 Deep Space 1 spacecraft The mass of the spacecraft is 474 kg and its initial velocity is 275 m/s. If a 56 mN forward force acts on the spacecraft during a displacement of 2.42×109 m. What is its final speed?

Use the work-energy theorem Wtotal = ΔKE to find the final velocity displacement displacement = 57 m vo = 3.6 m/s mass = 58 kg friction force = 71 N

Work done by gravity Only depends on the change in vertical position not on the path the object takes between the starting and ending points. Wg positive for downward motion

Example 7 Gymnast on a trampoline The gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast? Use the work-energy theorem Wtotal = ΔKE

Use the work-energy theorem Wtotal = ΔKE

Gravitational potential energy There is an easy way to calculate the work done by forces like gravity that at the same location always have the same magnitude and direction. PE is positive for heights above h = 0 PE is negative for heights below h = 0 h = 0

Conservative and non-conservative forces Forces like gravity and spring forces that can have a potential energy are called conservative forces. Conservative force rules: Work done is the same value for any path between the initial and final positions. Work done is zero for a closed path where the initial and final positions are the same. Forces like friction that do not follow these rules are called non-conservative forces.

Non-conservative forces Friction force is a non-conservative force because friction always acts backward for any motion. Work done by the kinetic friction force is always negative. For friction a longer path between the initial and final positions has more negative work. Friction fails rule 1. For friction a closed path the has the same initial and final positions will always have some negative work. Friction fails rule 2. Non-conservative forces can't have a potential energy.

Total work Total work done is equal to the work done by conservative forces plus the work done by non-conservative forces.

Define total mechanic energy E as E=KE+PE Total mechanical energy E Define total mechanic energy E as E=KE+PE

Conservation Laws Principle of conservation of total mechanical energy If the work done by non-conservative forces is zero, then the total mechanical energy does not change. Principle of conservation of total mechanical energy Principle of non-conservation of total mechanical energy If the work done by non-conservative forces is not zero, then the total mechanical energy does change.

Example: Bobsled sliding down hill with no friction Forces: gravity and support, Wnc = 0. Support force always does zero work. Why?

Example 8 Daredevil Motorcyclist Motorcyclist jumps across the canyon by driving off a cliff at 38 m/s. Find the velocity when she lands on the other side. Ignore air resistance. Use energy methods. mass = 200 kg Known: both positions, one velocity. Unknown velocity.

Example 8 Daredevil Motorcyclist Gravity is the only force that does work on the motorcycle while the motorcycle is in the air. (Ignore air resistance.) Gravity is a conservative force. Therefore, the work done by non-conservative forces is zero and the total mechanical energy remains constant. Ef = Ei Energy Initial Final Kinetic (KE) 144,400 J ? Potential (PE) 137,200 J 68,600 J Total (E) 281,600 J

Conceptual Example 9 Favorite swimming hole The person starts from rest with the rope held in the horizontal position, swings downward, and lets go of the rope. During the swing, two forces act on the person: gravity and the rope. If the initial and final heights are known, can the principle of conservation of energy be used to calculate the final velocity as the rope is released?

Example 11 Fireworks, non-conservative force Starting from rest, the force exerted by the burning rocket propellant does 425 J of non-conservative work. Find the final speed of the 0.2 kg rocket. Ignore air resistance.

Power Average power is the rate at which work is done or energy transferred, and is obtained by dividing the work or energy by the time interval.

Calculating the cost of using electric energy

Power example What average power is required to push a block at a velocity of 5 m/s with a 10 N force? 10 N 5 m/s In 1 second, the displacement "s" is 5 meters and θ = 0,

General principle of conservation of energy Energy can neither be created not destroyed, but can only be converted from one form to another. Energy conversion examples: Nuclear reactor Nuclear energy to thermal energy Engine Thermal energy to mechanical energy Electric generator Mechanical energy to electrical energy Electric motor Electrical energy to mechanical energy Mass is a form of stored energy. E = mc2 where "c" is the speed of light 300,000,000 m/s.

Work done by variable forces Work done by a constant force Work done by a variable force. Use lots of small displacements Δs. Work is equal to the area under the curve on an F vs. s graph. Or use the average force value

The End