PHYSICS | Custom PHD Thesis

PHYSICS

Historical Background

Galileo mathematically described the lever, as shown in Figure 10.1. It should be mentioned that

the lever was most probably identified first by Archimedes (287-212 B.C.). This is typical in the

history of science. Ideas develop through the centuries, and even millennia. Science is an

incredible collection and synthesis of ideas from all over the world and through out human

civilizations.

F fulcrum Rf = rF

r R

Figure 1. The lever.

The “R’s” are the distance from the fulcrum (lever arms), and the “F’s” are the forces at each end.

For a relatively small force, a large force can be produced. This idea is called leverage. A small

force gains leverage as its distance from the fulcrum increases.

Leverage is a ‘special case’ contained in Newtonian mechanics. The quantity, rF, is called

TORQUE. That is, T=rF. From Newton’s 2nd Law, it can be shown that if an object is not

rotating, then the sum of the Torques on that object must be zero, ?

T = 0.

In the next experiment we’ll learn about another definition of Torque that Newton defined. What

is significant about this other definition is that it explains the precession of the Earth, a truly

magnificent formulation for the 17th Century. Precession is usually attributed with gyroscopic

motion, which is governed by Torque.

Governing Principle

The expression Fr = fR was generalized by Newton. Newton showed this to be a direct result of

the principle of equilibrium. Newton stated that if a system was not accelerating, then the sum of

the external forces must be zero. Similarly, Newton stated that if a system was not angularly

accelerating, then the sum of the external torques must be zero.

Figure 10.2 shows a rigid body acted upon by several forces. If the system is not rotating, then

the sum of the torques caused by the forces must be zero. That is, F1r1 – F2r2 – F3r3 = 0. A

positive sign is assumed for counterclockwise rotation and a negative sign for clockwise rotation.

F2 F3

Figure 10.2. Rigid body under several torques.

Experimental Procedure

For the problems, use the following diagrams and equations. Use a ruler clamp to hang and

balance a meter stick from a ring stand.

r1 r2

m1r1 = m 2 r2

?T = 0:

m3 r1m1 g – r2 m 2 g – r3 m3 g = 0

m1 m2

r1 x

mcm

m1r1 = mcm x

Experimental Data: Experiment 9

For the given values, determine the unknown quantities. Verify the results experimentally by

constructing the configurations. Show all calculations explicitly and neatly.

Case 1: r1 = 5 cm, m1 = 500 g, m2 = 100 g.

r2 = _________(calculated)

r2 = _________(experimental)

Case 2: r1 = 20 cm, r2 = 30 cm, m1 = 500 g, m2 = 200 g, m3 = 100 g.

r3 = _________(calculated)

r3 = _________(experimental)

Case 3: r1 = 25 cm, r2 = 10 cm, m1 = 500 g, m2 = 250 g, r3 = 20 cm.

m3 = _________(calculated)

m3 = _________(experimental)

Clamps were used to hold the weights in this experiment. If each clamp weighed 17 g, how

would this affect your answers in Cases 1 and 2?

r2 = _________(Case 1)

r3 = _________(Case 2)

PHY 101 LAB NAME___________________________

Torque & Equilibrium

Case 4: r1 = 10 cm, m1 = 100gm + weight clamp .

(you have to use trial and error to find x, start with x about 8 cm)

x = _________(experimental)

mcm =________(calculated from data)

Case 5: x = 10 cm, m1 = 100gm + weight clamp .

r1 = _________(experimental)

mcm = ________(calculated from data)

Calculate the average mass of the meter stick from Cases 4 and 5. Measure the mass of the meter

stick with an electronic scale and calculate the percent error. Use the electronic scale reading as

an accepted value.

mcm = _________(average)

m cm =_________(scale)

% error = _________

Questions: Experiment 9

Answer each question completely with explanations of all answers. Show all calculations.

1. A 200 lb man walks out to the end of a uniform horizontal plank which projects

perpendicularly over the edge of a roof. The plank is 20 feet long and weighs 100 lb. How

far from the edge of the roof can the plank overhang?

2. A 10,000 kg truck cruises over a 30 m span bridge. What are the normal forces on the ends

of the bridge when the truck is 10 m from the right side, if the mass of the bridge itself is

ignored?

3. A uniform ladder 4 m long rests at 45-degree angle against a frictionless wall. If the ladder

has a mass of 10 kg, what is the frictional force at the base of the ladder? (Hint: make a free

body diagram and consider both trans. and rot. equilibrium.)

4. A uniform beam, of length L and m = 2.2 kg, is at rest with its ends on two scales. A

uniform block, with mass M = 3.6 kg, is at rest on the beam, with its center a distance,

from the beam’s left end. What do the scales read?

5. A uniform meter stick supported at the 25 cm mark is in equilibrium when a 200-gm rock is

suspended at the 0 cm end. Is the mass of the meter stick greater than, equal to, or less than

the mass of the rock? Explain your reasoning.

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