Energy & Work (Cont.)

4 Basic Simple Machines Levers Inclined planes Pulleys Wheel and axle

How much easier is it to do the work (1470J) with the inclined plane than without? 3 m5 m Mechanical advantage = force without incline / force with incline 490 N 294 N Force w/o Force w = =1.66 For inclined planes you get the same number by: Length of plane Height of plane = 5 m 3 m = 1.66

1 st Class Lever Fulcrum Effort arm dede FeFe drdr FrFr Resistance arm

The work done on one side of the lever must equal or exceed the work performed by the other side. F e *d e ≥ F r *d r Fulcrum Effort arm dede FeFe drdr FrFr Resistance arm

If we have a 150 kg mass and we need to life it 1 m, and we have a 1 st class lever that is 3 m on the other end, what is the minimum force necessary to lift the weight? FeFe F r 150 kg 3m de 1m dr W = F*d F = m*a Recall : W e ≥ W r F = W/d F = (1470 kg*m 2 /s 2 )(3 m) = (150 kg)(9.8 m/s 2 )= 1470 kg m / s 2 = (1470 kg m / s2)(1m)= 1470 kg m2 / s2= 1470 Joules (J) = 490 kg*m/s2 = 490 N

Alternative way to look at the problem: For all simple machines: W in = W out But W= F*d, so:F in *d in = F out *d out in = effort out = resistance If you know any 3 of these, you can solve for the 4th But F out =m*a; F out *d out d in F in = m*a*d out d in F in = (150 kg*9.8m/s 2 )(1 m) 3 m = = 490 kg*m/s 2 = 490 N

Mechanical Advantage It is a measure of how much easier or harder it is to do work with a machine F out / F in In the previous example = 1470 N / 490 N = 3 For levers, a quick way to determine mechanical advantage is:  MA = Effort arm / Resistance arm

What is the mechanical advantage of the lever system? E R d e 5 kg 15 kg F out F in 15 kg * 9.8 m / s 2 = 5 kg * 9.8 m / s 2 in = effort out = resistance

2 nd Class Lever Fulcrum Effort arm dede FeFe drdr FrFr Resistance arm

3 rd Class Lever Fulcrum Effort arm dede FeFe drdr FrFr Resistance arm

Humerus Ulna Radius Olecranon Process

If I am lifting a 25 pound weight with my biceps, what is the mechanical advantage, and how much force (minimum) must the biceps apply? Fulcrum Effort arm dede FeFe drdr FrFr Resistance arm It is about 5 cm from elbow to point of attachment of muscle and the whole bone is about 33 cm What do we need to know? Recall: mechanical advantage is effort arm/resist. arm Therefore, 5 cm/ 3 cm = 0.15 Recall W e = W r F e d e = F r d r F e = F r d r /d e F e = 25lb.*33cm / 5 cm = 165 lbs.

Pulleys Fixed pulleys only change the direction of a force They provide no mechanical advantage W=F*d

Moveable pulleys supply mechanical advantage to the number of lines attached to the mass W=F* d If the box weights 100 kg, how much force is needed to lift it without the pulley? F = ma= (100 kg)(9.8 m/s 2 )= 980 N How much force would you have to use using the pulley? For all simple machines: W in = W out But W= F*d, so:F in *d in = F out *d out 980N*d = F*2*d 980N / 2 = F 490 N = FF = 490 N

The Wheel and Axle As in all simple machine, Fd = Fd Circumference of axle = d = 2πr axle Circumference of wheel = d = 2πr wheel Therefore, F axle r axle = F wheel r wheel A small force on the wheel translates into a large force on the axle A slow velocity on the axle becomes a fast velocity on the wheel r axle r wheel Wheel Axle