And if the force is constant, we have our old friend... W = Fd cos θ
A note on the sign of work When 0º ≤ θ < 90º work is positive When θ = 90º work is zero When 90º < θ ≤ 180º work is negative
Ex. A book with a mass of 2.0 kg is lifted at a constant velocity to a height of 3.0 m. How much work is done on the book?
Ex. A 15 kg crate is moved along a horizontal surface by a warehouse worker who is pulling on it with a rope that makes an angle of 30.0° with the horizontal. The tension in the rope is 100.0 N and the crate slides a distance of 10.0 m. a) How much work is done on the crate by the rope? b) How much work is done on the crate by the normal force? c) How much work is done on the crate by the frictional force if the coefficient of kinetic friction is 0.40?
Ex. A box slides down an inclined plane that makes an angle of 37° with the horizontal. The mass of the block is 35 kg, the coefficient of friction between the box and ramp is 0.30 and the length of the ramp is 8.0 m. How much work is done by a) gravity? b) the normal force? c) friction? d) What is the total work done?
Conservation of Mechanical Energy (makes lots of problems simple that might otherwise be horrendous)
Ex. A child of mass m starts from rest at the top of a water slide 8.5 m above the bottom of the slide. Assuming the slide is frictionless, what is the child’s speed at the bottom of the slide?
Note: if no non-conservative forces (friction, tension, air resistance) are present, the change in energy is independent of the path taken. Gravity is a conservative force.
Ex. A box, with an initial speed of 3.0 m/s, slides up a frictionless ramp that makes an angle of 37° with the horizontal. How high up the ramp will the box slide? What distance along the ramp will it slide?
Ex. A skydiver jumps fro a hovering helicopter that is 3000 m above the ground. If air resistance is negligible, how fast will the skydiver be falling when his altitude is 2000 m?
Modifying conservation of energy to include non- conservative forces
Ex. Wile E. Coyote (mass = 40 kg) falls off a 50 m high cliff. On the way down, the average force of air resistance is 100 N. Find the speed with which he crashes into the ground.
Ex. A skier starts from rest at the top of a 20.0° incline and skis in a straight line to the bottom of the slope a distance d (measured along the slope) of 400 m. If the coefficient of kinetic friction between the skis and the snow is 0.20, calculate the skier’s speed at the bottom of the run.
Key aspects of a potential energy curve The object moves in a linear path along + x and –x-axis The force is the negative slope of the curve E mec = U(x) + K(x) Turning points: locations where the object is at rest; i.e. K(x) = 0 and E mec = U(x) Object is in equilibrium (neutral, unstable, stable) when slope = 0 (F(x) = 0) Object cannot reach locations where K is negative To graph: graph zeros (F(x) = 0), regions of constant force and connect.
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