36 Unit 1 Motion and force

ChAPTER 3 • Forces

Investigation 3.3Levers and Rotational EquilibriumHow do levers work?The development of the technology that created computers, cars, and the space shuttle began with the invention of simple machines. A simple machine is a mechanical device that does not have a source of power and accomplishes a task with only one movement. The lever is an example of a simple machine. A lever allows you to move a rock that weighs 10 times as much as you do (or more)!

 How do levers work?A lever includes a stiff structure (the lever) that rotates around a fi xed point called the fulcrum. The side of the lever where the input force is applied is called the input arm. The output arm is the end of the lever that moves the rock or lifts the heavy weight.

Levers use torque to lift or move objects. Torque is a force applied over a distance that causes rotation to occur. In a lever, the force is applied perpendicular to the distance. This perpendicular force causes rotation to occur. To calculate torque, you multiply the force applied times the distance of the force from the fulcrum. Torque is measured in units of force (newtons) times distance (meters), or newton-meters (N-m). The example above shows how to calculate torque.

4Levers set

4Physics Stand

4One set of weights

4Spring scales capable of measuring forces from 0 to 10 N

420 glass marbles (from Atom Building Game)

Input arm

Output armFulcrum

Lever

Input force

Output force

Force = 100 N

Hinge (fulcrum)

Distance = 1.0 m

CALCULATE TORQUE

torque = force × distance = 100 N × 1.0 m = 100 N-m

MATERIALSLIST

37Investigation 3.3 levers and Rotational Equilibrium

ChAPTER 3 • Forces

Use the example to the right to answer the following questions.

a. Calculate torque in the example.b. Based on your answer, why are doorknobs placed

as far away from the hinge as possible?c. What would happen if you applied the same

amount of force directly to the hinge? (Hint: Find the torque.)

 Levers and torque1. Set up the lever as shown in the diagram below. You can secure the mass to the lever by looping the

string around the distance indicating post on the lever for this part of the investigation, as shown in the diagram below. This way, you will know the exact distance of the mass from the fulcrum.

2. Use a spring scale to measure the force exerted by a weight.3. Calculate the input torque by multiplying the amount of force on the input arm by the distance from

the fulcrum.4. Calculate the output torque by multiplying the amount of force on the output arm by the distance from

the fulcrum. Show this as a negative value because it causes the lever to rotate in the opposite direction as the input torque.

5. Conduct more trials by hanging different combinations of weights on both sides of the fulcrum so that the lever balances. Record your results in the chart on the next page. There is room for six trials.

100 N

Hinge

0.5 m

1. Setup 2. Measureforce

3 & 4. Calculate input and output torques

5. Try other combinations that balance the lever

Output torque = (1 N × 0.1 m) + (0.33 N × 0.3 m) = 0.1 N-m + 0.99 N-m = –1.99 N-m

Input torque = 1 N × 0.2 m = + 0.2 N-m

1 N

0.1 m

1 N 0.33 N

0.3 m

Output arm

WeightsWeights

Fulcrum

Input arm

0.2 m

Output arm

WeightsWeights

FulcrumFulcrum

Input arm

38 Unit 1 Motion and force

ChAPTER 3 • Forces

Data Chart

 What did you learn?a. How do the input and output torques compare for each trial?b. Explain, using your knowledge of torque, why the lever is in balance.c. Use your data to identify the mathematical relationship between input force, input length, output force,

and output length. Write the relationship as a mathematical formula.d. Why do you think the term rotational equilibrium is used to describe when the lever balances?

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

The distance in-between the strings is the distance from the fulcrum. In this case, it is 22 cm.

Clip attaches mass to top of lever

39Investigation 3.3 levers and Rotational Equilibrium

ChAPTER 3 • Forces

Data Chart

 What did you learn?a. How do the input and output torques compare for each trial?b. Explain, using your knowledge of torque, why the lever is in balance.c. Use your data to identify the mathematical relationship between input force, input length, output force,

and output length. Write the relationship as a mathematical formula.d. Why do you think the term rotational equilibrium is used to describe when the lever balances?

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

Input torque30 cm 20 cm 10 cm 30 cm20 cm10 cm

Output torque

5cm 5cm10 1015 1520 2025 2530 30

Output armInput arm

The distance in-between the strings is the distance from the fulcrum. In this case, it is 22 cm.

Clip attaches mass to top of lever

 Using the lever to solve a mysteryYou have seen what it takes to make the lever reach equilibrium and be in balance. Now it is up to you to use this condition, and what you have learned about the mathematical relationship between both sides of the lever to uncover the mass of one single glass marble. You may consider using the back mass hanging clips for this part. You can move hanging masses to any location on the lever using these, and they will clearly indicate their exact distance from the fulcrum.

a. What condition had to be in place for the lever to balance?b. You have 20 glass marbles and a small bag that can be used to

hang them from the lever. (1) Describe a process that you could follow using the lever and weights to determine the mass of one single marble. (2) Explain what each step would be, why it is being done, and be sure to include terms like input torque, output torque, equilibrium, force, and distance. (3) The mass of one single marble must be given in grams, so include how you would determine that from what you know about the relationship between force and mass.

c. Once you have completed your plan, use it to determine the mass of one single marble using the lever and weights. What was your result?

d. How did your result compare to the other groups in your class?e. Use an electronic scale or a triple beam balance to determine the mass of one single marble. How does

this compare to your result?f. Is getting the mass of one single marble the best way to evaluate your result? What might be a better

method? (Hint: Do you think all of the marbles have the exact same mass?)g. Use your method and compare your result to this new fi gure. Are you satisfi ed with your result?h. If time permits, discuss with your class all of the things you could have done to refi ne your experiment,

try these methods, and see if you can get a closer result. Was your result much closer?