Honors Physics Unit 9 Review: Work, Energy, and Simple Machines

Work is energy transferred by a force acting through a displacement. For a constant force, $W = Fd\cos\theta$, where $\theta$ is the angle between the force and the displacement. A force along the motion does positive work. A force opposite the motion, like kinetic friction, does negative work. A force perpendicular to the motion does zero work.

The work-energy theorem links work to speed. The net work done by all forces equals the change in kinetic energy, $W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$. Positive net work speeds an object up, and negative net work slows it down.

Power measures how fast work gets done. Average power is $P_{avg} = W/t$, measured in watts (1 W = 1 J/s). When the force points along the velocity, you can also use $P = Fv$.

Worked example. A 2.0 kg box starts at rest. You push it 5.0 m with a 10 N horizontal force while friction pulls back with 4.0 N. Your work is $(10)(5.0) = 50$ J. Friction's work is $(4.0)(5.0)\cos 180^\circ = -20$ J, so $W_{net} = 30$ J. Then $30 = \frac{1}{2}(2.0)v_f^2$, which gives $v_f^2 = 30$ and $v_f \approx 5.5$ m/s. The push took about 1.83 s, so your average power output was $50/1.83 \approx 27$ W.

Mechanical energy is kinetic plus potential energy, $ME = KE + PE$. Kinetic energy is $\frac{1}{2}mv^2$. Near Earth's surface, gravitational potential energy is $mgh$, measured from a reference height you choose. Elastic potential energy in an ideal spring is $\frac{1}{2}kx^2$, where $x$ is the stretch or compression from the spring's natural length.

Energy is never created or destroyed. When only conservative forces like gravity and springs do work, mechanical energy stays constant, so $KE_i + PE_i = KE_f + PE_f$. A pendulum or a frictionless roller coaster trades PE for KE and back without losing any.

Friction and air resistance often reduce mechanical energy, converting some into thermal energy and sound. Track nonconservative work with $KE_i + PE_i + W_{nc} = KE_f + PE_f$. The work $W_{nc}$ can be negative or positive, depending on how the force acts.

Worked example. A 50 kg skateboarder starts from rest at the top of a 5.0 m ramp. Ignoring friction, $mgh = \frac{1}{2}mv^2$, so the mass cancels and $v = \sqrt{2gh} = \sqrt{2(9.8)(5.0)} \approx 9.9$ m/s at the bottom. Suppose she actually reaches the bottom at 8.0 m/s. Her starting energy was $(50)(9.8)(5.0) = 2450$ J and her final KE is $\frac{1}{2}(50)(8.0)^2 = 1600$ J. Friction converted the difference, 850 J, into thermal energy.

A simple machine changes the size or direction of a force. For a passive machine, useful work out cannot exceed work in. With constant forces along their displacements, work in is $F_{in}d_{in}$, and an ideal machine gives back the same amount as work out. If the input force is smaller than the output force, the input distance is longer.

Ideal mechanical advantage (IMA) comes from the machine's geometry, usually $d_{in}/d_{out}$. Actual mechanical advantage (AMA) comes from measured forces, $F_{out}/F_{in}$. Friction makes AMA smaller than IMA. Efficiency is $\frac{W_{out}}{W_{in}} \times 100\%$. For constant forces along the input and output displacements, it also equals $\frac{AMA}{IMA} \times 100\%$.

The six simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw. A single fixed pulley has an IMA of 1, so it only changes the direction of your pull. Third-class levers like tweezers or a fishing rod have a mechanical advantage below 1, which means they give up force to gain distance and speed at the output.

Worked example. You push a 600 N crate up a 4.0 m ramp onto a 1.0 m platform. The IMA is $4.0/1.0 = 4$, so a frictionless ramp would need only 150 N. You actually need 200 N, so the AMA is $600/200 = 3$. Work out is $(600)(1.0) = 600$ J and work in is $(200)(4.0) = 800$ J, so the efficiency is $600/800 = 75\%$, which matches $3/4$.