Torque and Simple Machines Physics Section 8.2 Torque and Simple Machines

Torque Torque is the rotational analog to linear force. A net torque produces an angular acceleration. Torque is the tendency of a force to rotate an object around a axis, fulcrum, or pivot. The symbol for torque is τ (tau). The units for torque are newton•meters (Nm).

× τ = r × F τ = r F τ = r F sinθ F F Magnitude of torque is given by: causes rotation F τ = r F F τ = r F sinθ pivot doesn’t cause rotation lever arm torque up, out of paper Direction of the torque vector: × up, out of paper down, into paper

The grip right hand rule Net Torque Στ = τcw + τccw Net torque is the sum of clockwise and counterclockwise torques. When there is no angular acceleration, Στ = 0. This is called rotational equilibrium.

Simple Machines A simple machine is a non-motorized mechanical device that changes the direction or magnitude of a force to create a mechanical advantage. The main simple machines are: • Lever • Wheel and axle (includes gears) • Pulley • Inclined plane • Wedge Screw Simple machines are the building blocks of more complex machines. For example, a bicycle is a machine that consists of levers, wheel and axles, and pulleys.

Mechanical Advantage An ideal machine has no loss of energy due to friction (waste heat). In an ideal machine, the work put into the machine, equals the work out. Win = Wout therefore, Fin din = Fout dout where Fin is the force put into the machine, din is the displacement over which that force acts, Fout is the force on the load, and dout is the displacement of the load (amount a load is lifted, etc.). Since the work done on the ideal simple machine and by the simple machine is done during the same time interval, t, we can also say power in is equal to power out: Pin = Pout therefore, Fin vin = Fout vout where vin and vout are the velocities of the input and output parts of the machine. The mechanical advantage of the machine is the ratio of the force out to the force put in: MA = Fout / Fin Usually, the force in is called the effort force, or effort. The force out is called the load or resistance.

Ideal Mechanical Advantage In the ideal machine, Fin din = Fout dout or Fe d e = Fr d r where e is the effort, and r is the resistance or load. We can rearrange this to give: This ratio is called the ideal mechanical advantage, or IMA. Note that force and displacement are reciprocal. This shows that one way to define a machine is that it reduces the force by increasing the displacement over which the force acts. Fr Fe = de dr de IMA = dr

Actual Mechanical Advantage Real machines do lose some energy from friction. You will not get as much force out as you might expect for the ideal machine. Real machines are not 100% efficient. If the effort and load forces are measured, the ratio gives the actual (real) mechanical advantage, AMA. AMA = Fr / Fe Efficiency The efficiency of the machine, ℯ, is a measure of the actual work out compared to what is put in. Since some energy is lost, the work out will not equal the work in. ℯ = Wout / Win = Fr d r / Fe d e = (Fr / Fe)(dr / de) So, ℯ = (AMA / IMA) × 100

Types of Simple Machines Lever There are three classes of lever, class 1, class 2, and class 3. The class of lever depends on the position of applied effort, the fulcrum (pivot point), and the load. The lever arm from the fulcrum to the effort force is the effort arm. The lever arm from the fulcrum to the load is the load or resistance arm. The distance each moves through an arc is the same ratio as the length of these two lever arms. Therefore, for a lever: IMA = effort arm length ÷ load arm length dE ℓL ℓE dL

Class 1 — fulcrum in the middle Examples: crow bar, can opener, bicycle hand brakes, claw on hammer for pulling out nails. Class 2 — resistance (load) in the middle Mechanical advantage is always more than 1. Examples: wheel barrow, nut cracker, bottle opener.

Class 3 — effort in the middle Mechanical advantage is always less than 1. Examples: tweezers, human jaw (mandible), human forearm lifting a weight, fishing rod.

Wheel and Axle The wheel and axle works on the same principle as the lever, but uses a wheel as for the applied effort force, and an axle for lifting or turning a load. A rotating handle may act as the wheel. IMA = radius of wheel ÷ radius of axle. Examples: door knob, lug wrench, gears.

Gears Gears are a type of wheel and axle that have teeth on the edges of wheels and axles. IMA = number of teeth on output gear ÷ number of teeth on input gear IMA = Nout ÷ Nin

Pulley A pulley system uses a rope or cable, and one or more rotating grooved wheels, the pulley. A set of pulleys assembled so that they rotate independently on the same axle form a block. Two blocks with a rope attached to one of the blocks and threaded through the two sets of pulleys form a block and tackle. The mechanical advantage is equal to the number of sections (strands) of the rope actually holding up the load. This is because the weight of the load is divided among all the ropes supporting the load. IMA = ropes supporting load

IMA on the these tackles are 2, 3, 4, 5 and 6. Double Tackle: 100 N is lifted by 25 N.

Inclined Plane A ramp or inclined plane is used to lift an object up a vertical distance by moving it along a longer diagonal path, thus trading longer displacement for smaller force. IMA = diagonal ramp length ÷ rise

Wedge A wedge works as a double inclined plane. It pushes straight apart by forcing the wedged material along two inclines. All cutting and chopping blades are wedges. IMA = slant length ÷ width of wedge

Screw A screw is a form of inclined plane in the form of a spiral. The effort is in turning the screw through its full circumference, π d. The load is lifted through a length, ℓ. IMA = π d ÷ ℓ where d is the diameter of the screw or bolt shaft. The length is often given as the pitch or thread height of the screw, since that is the length the screw is moved when rotated once. To find the pitch, find the inverse of the threads per centimeter or inch. Pitch = 1 ÷ thread count per unit length Examples: wood screw, sheet metal screw, machine bolt, automobile screw jack.