Name:
Date:
3/29/2023
Student Exploration: Torque and Moment of Inertia
Directions: Follow the instructions to go through the simulation. Respond to the questions and
prompts in the orange boxes.
Vocabulary: angular acceleration, fulcrum, lever, moment of inertia, Newton’s second law, torque, weight
Prior Knowledge Question (Do this BEFORE using the Gizmo.)
During recess, Tom and his little sister Marcie want to play on the see-saw. Tom is quite a bit heavier than
Marcie. Where should they sit so the see-saw is balanced? Sketch their positions on the image below.
Explain your reasoning:
Since Tom weighs more, he should sit
closer to the center of the see-saw in
order to balance out the weight. Marcie
can stay where she is.
Gizmo Warm-up
The Torque and Moment of Inertia Gizmo shows a see-saw,
which is a type of lever. The see-saw can hold up to eight
objects. To begin, check that the Number of objects is 2. Check
that the mass of object A is 1.0 kg and the mass of object B is
2.0 kg. The two objects are equidistant from the triangular
fulcrum that supports the lever.
1. Click Release. What happens?
The see-saw tilts towards the right since the object B is heavier.
2. Click Reset. Without changing the masses, experiment with different positions of objects A and B by
dragging them around.
Can you create a scenario in which object A goes down and object B goes up?
Explain:
When object B is closer to the center of the see-saw, and object A is closer to
the end of the see-saw, object A goes down and object B goes up.
3. Can you create a scenario in which object A perfectly balances object B?
Explain:
Yes
Yes
When object B is 2x as far as object A. For example, when object A is 1.0m
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from the center and object B is 0.5m from the center.
Activity A:
Get the Gizmo ready:
● Click Reset. Turn on Show ruler.
● Check that object A is 1.0 kg and B is 2.0 kg.
Principle of the
lever
Question: How can you use a light object to balance a heavy object?
1. Explore: Experiment with the Gizmo to see how you can balance a heavy object with a light object. What
do you notice about the distances of each object from the fulcrum?
When you make the heavier object closer to the center, and the lighter object further from the center,
both objects are able to balance. In order to balance objects of different weights, the distance of the
lighter object must be double the distance of the lighter object. (When the weights are different by a
1:2 ratio)
2. Gather data: For each mass and location of object A, find a location for object B so it perfectly balances
object A. You can change the mass of object A by typing the mass into the text box and hitting “Enter” on
your keyboard. Leave the mass of object B the same
(1 kg) in each experiment. Include all units in the table.
Object A
mass
Object A
location
Object B
mass
Object B
location
Object A
m×d
Object B
m×d
1.0 kg
-0.4 m
1.0 kg
0.4 m
0.4
0.4
2.0 kg
-0.4 m
1.0 kg
0.8 m
0.8
0.8
3.0 kg
-0.4 m
1.0 kg
1.2 m
1.2
1.2
4.0 kg
-0.4 m
1.0 kg
1.6 m
1.6
1.6
3. Analyze: What patterns do you notice in your data?
I notice that if object B stays the same, and object A changes, the location of object B changes by
how much object A’s mass changes.
4. Calculate: Fill in the last two columns by multiplying each object’s mass by its distance from the fulcrum.
The units are kg·m. (Note: The distance d is always a positive number.)
What do you notice?
They are the same.
5. Generalize: In general, how can you calculate the distance of object B from the fulcrum so that it balances
object A?
To calculate the distance of object B from the fulcrum, you multiply the mass and location of object A,
and divide by the mass of object B.
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6. Apply: Suppose you wanted to lift a heavy rock with a lever. Would you place the fulcrum near the rock or
near the part of the lever where you are pushing? Explain.
If you want to lift a heavy rock with a lever, you would place the fulcrum near the part you are pushing
because it will be easier to lift. If it was further from where you are pushing, then it would be harder to
push, as seen when you put the heavier object farther from the center of a see-saw, and the lighter
object closer to the center. The heavier object would fall since it is so far from the center, thus, it would
be easier to lift if it's closer to the center.
Activity B:
Torque
Get the Gizmo ready:
● Click Reset. Turn on Show initial torque.
● Set the Number of objects to 1.
● Set Mass A to 2.0 kg.
Question: What is the rotational force that an object exerts on a lever?
1. Calculate: When object A is positioned on the see-saw, it is pulled down by the force of gravity. The
gravitational force on an object, or its weight (w), is equal to its mass multiplied by gravitational
acceleration (g). Gravitational acceleration is 9.81 m/s2 on Earth’s surface.
What is the weight of object A?
19.62 N
[Note: The unit for weight is the newton (N).]
2. Predict: The twisting force an object exerts on a see-saw is called torque (τ). How do you think the torque
depends on the distance of object A from the fulcrum?
The torque decreases when the distance from the fulcrum decreases.
3. Gather data: Place object A at several locations on the see-saw, on both sides of the fulcrum. Use a
different mass in each experiment. In each trial, click Release and record the initial torque. Record object
A’s mass, weight, location, and torque in the table below.
Object A mass
(kg)
Object A weight
(N)
Object A location (m)
Object A torque
(N·m)
0.5 kg
4.91 N
- 0.5 m
2.45 Nm
1.0 kg
9.81 N
- 1.0 m
9.81 Nm
3.0 kg
29.43 N
- 1.5 m
44.15 Nm
7.0 kg
68.67 N
- 2.0 m
137.34 Nm
10.0 kg
98.1 N
- 2.0 m
196.20 Nm
4. Analyze: Based on your data, write an equation for torque. Use the symbol r to represent distance. For
now, ignore the sign of the torque. Test your equation with the Gizmo.
τ = mgr
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5. Make a rule: Now focus on the sign of each torque value in your table. How does the sign relate to the
direction of rotation? (Fill in each blank with “clockwise” or “counterclockwise.”)
If torque is positive, the resulting motion is clockwise
If torque is negative, the resulting motion is counterclockwise
6. Apply: What is the torque exerted by a 4.2-kg mass that is located 1.8 m to the right of the fulcrum?
(4.2)(9.81)(1.8)= -74.16 Nm
Check your answer with the Gizmo.
7. Explore: Set the Number of objects to 2. Set the Mass of object A to 5 kg and its Location to 1.2 m. Set
the Mass of object B to 3.0 kg and is Location to 0.5 m.
A. What torque does object A exert on the see-saw?
(5)(9.81)(1.2) = -58.86 Nm
B. What torque does object B exert on the see-saw?
(3)(9.81)(0.5) = -14.715 Nm
C. What do you think is the total torque on the see-saw?
- 73.575 Nm
D. Check that Show initial torque is on and click Release.
What is the total torque?
- 73.58 Nm
8. Practice: A lever supports four objects:
●
Object A is 3.0 kg and located 2.0 m left
of the fulcrum.
●
Object C is 8.0 kg and located 0.1 m
right of the fulcrum.
●
Object B is 7.0 kg and located 0.5 m left
of the fulcrum.
●
Object D is 4.5 kg and located 1.6 m
right of the fulcrum.
A. What is the total torque on the lever?
14.715 Nm
(Hint: Recall that objects to the right of the fulcrum will have a negative torque.)
Show your work:
2
2
[− (3. 0𝑘𝑔)(9. 81𝑚/𝑠 )(− 2. 0𝑚)] + [− (7. 0𝑘𝑔)(9. 81𝑚/𝑠 )(− 0. 5𝑚)]
2
2
+ [− (8. 0𝑘𝑔)(9. 81𝑚/𝑠 )(0. 1𝑚)] + [− (4. 5𝑘𝑔)(9. 81𝑚/𝑠 )(1. 6𝑚)] =
(58. 86) + (34. 335) + (− 7. 848) + (− 70. 632) = 14. 715 𝑁𝑚
B. When released, will the left rotate clockwise or counterclockwise?
clockwise
C. Check your answers on the Gizmo. Were you correct?
Yes
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9. Explain: If two kids are playing on the see-saw, why should the larger kid sit closer to the fulcrum than the
smaller kid? Use the term “torque” in your explanation.
The larger kid should sit closer to the fulcrum since he will exert less torque. If they both sit an equal
distance away, the see-saw will always make the lighter kid go up. However, when you play on a
see-saw you want to have the children balance the see-saw so they can use their legs to push up and
down on each other.
Activity C:
Moment of inertia
Get the Gizmo ready:
● Click Reset.
● Check that the Number of objects is 1.
● Set Mass A to 2.0 kg.
Introduction: When describing the motion of rotating objects (such as see-saws), physicists use several terms
that are equivalent to those used for linear motion. For example, torque (τ) is the rotational equivalent of force,
while angular acceleration (α) is the rotational equivalent of linear acceleration.
Question: What factors affect how quickly a see-saw accelerates?
1. Predict: Two children of equal mass decide to have a see-saw race. Each child sits on an identical see-saw
with nothing on the other side. William sits at the end of his see-saw as far away as possible from the
fulcrum. Kate sits near the middle of her see-saw close to the fulcrum. Their friends lift both see-saws to
the top and release them simultaneously.
Which see-saw do you think will hit the ground first, and why?
William will hit the ground first because he is a further distance from the fulcrum. Since they both
weigh the same, only distance will affect a difference in torque. To increase torque, you increase the
distance, so when William sits further, he will release more torque on the see-saw than Kate.
2. Experiment: Move object A to the end of the see-saw as far as possible from the fulcrum. Click Release,
and record how long it takes for the see-saw to hit the ground. Click Reset and place object A close to the
fulcrum. Click Release and record the time again.
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Time to hit ground when object A is far from the fulcrum:
0.4 s
Time to hit ground when object A is close to the fulcrum:
0.09 s
3. Compare: The rate at which a rotating object accelerates is related to its moment of inertia. Click Reset.
Turn on Show moment of inertia and compare the moment of inertia for a mass close to the fulcrum and
the same mass far from the fulcrum. (Click Release to see the moment of inertia.) How does the moment
of inertia for the close mass compare to that for the farther mass?
The moment of inertia for a mass closer to the fulcrum is a lot smaller than the moment of inertia for
2
a mass far from the fulcrum. The close mass’s moment of inertia is 0.02 𝑘𝑔 · 𝑚 , while the further
2
mass’s moment of inertia is 8.00 𝑘𝑔 · 𝑚 .
4. Calculate: Place a 5.0-kg mass 2.0 m from the fulcrum. To find the moment of inertia for a mass located a
distance r from the fulcrum, use the equation: I = mr2.
What is the moment of inertia of this mass?
2
20. 0𝑘𝑔 · 𝑚
(Note: Units are kg·m2.)
Check your answer by turning on Show moment of inertia and clicking Release.
5. Compare: You may have been surprised that the see-saw accelerated more slowly when the mass was far
from the fulcrum and the torque was greater. That is because the angular acceleration of the see-saw
depends on two factors: torque and moment of inertia (I). Just as mass is a measure of an object’s
resistance to acceleration, moment of inertia is a measure of an object’s resistance to angular acceleration.
Compare these two equations:
Newton’s second law (linear motion)
F = ma
Newton’s second law (rotational motion)
τ = Iα
How are these equations similar?
While the linear motion equation measures linear acceleration and mass, the measure of an
object's resistance to acceleration, rotational motion measures the same thing, just in terms of
angular acceleration. The linear motion equation is just the measure of an object’s resistance to
linear acceleration times linear acceleration. The rotational motion equation is just the measure of
an object’s resistance to angular acceleration times angular acceleration.
6. Manipulate: Start with the rotational version of Newton’s second law (τ = Iα).
A. Solve this equation for the angular acceleration.
α=
B. Substitute mr2 for I in this equation.
α=
τ
𝐼
τ
2
𝑚𝑟
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C. Recall that torque is defined as force multiplied by radius.
Substitute Fr for τ into your equation.
α=
D. For an object on the see-saw, the force on the object is equal
to its weight, mg. Substitute mg for F in your equation
α=
E. Cancel everything that can be canceled to find a simplified
expression for the angular acceleration of the see-saw.
α=
𝐹𝑟
2
𝑚𝑟
𝑚𝑔𝑟
2
𝑚𝑟
𝑔
𝑟
7. Interpret: Look at your expression for the angular acceleration. (If possible, check this equation with your
teacher.)
A. Does the angular acceleration depend
on the mass of the object?
Explain:
B. How does the angular acceleration
change as the distance (r) increases?
No, it does not depend on the mass of the object.
It does not depend on the mass of the object, since
it cancels out. It only depends on gravity and the
distance.
As the distance increases, the acceleration of the
object decreases since r is in the denominator.
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