5.9 - Numerical Problems on Moments
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Introduction to Moments with Practical Examples
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Today we're going to explore how to calculate moments using practical examples. Can anyone remind us how we define a moment?
Is it the turning effect produced by a force applied at a distance from a pivot?
Exactly! We denote it with the formula M = F × d. Let's use this to calculate a moment. If we have a force of 20 N applied at a distance of 4 m from the pivot, what would the moment be?
We would multiply 20 by 4, which gives us 80 Nm.
Correct! Great job, everyone. This means we have an 80 Nm moment from that force.
Applying the Principle of Moments
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Now let's apply the principle of moments. Who can tell me what that principle states?
It says that the sum of clockwise moments equals the sum of anticlockwise moments when a body is in equilibrium.
Well said! Let’s say we have a 30 N force applied 2 m from the left support, and a 50 N force applied 3 m from the same support. How would we set up our equation?
We can write it as 30 × 2 = 50 × x, where x is the distance from the left support to the pivot.
Exactly! So, what do we find if we solve for x?
x = 1.2 m.
Great work! This shows how balancing forces allow us to calculate unknowns using moments.
Introduction & Overview
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Quick Overview
Standard
In this section, numerical examples illustrate how to calculate the moment produced by a force and how to apply the principle of moments in various scenarios, including determining unknown distances in beams under load.
Detailed
Detailed Summary
This section delves into practical numerical problems that illustrate the concept of moments, focusing on how to compute moments generated by forces and how to use the principle of moments to solve various engineering problems. It introduces two distinct examples:
Example 1: Moment of a Force
A scenario where a force of 20 N is applied at a distance of 4 meters from the pivot is presented. The calculation of the moment is straightforward and employs the formula:
$$M = F \times d$$
By substituting the values:
- F = 20 N
- d = 4 m
The resulting moment is calculated as:
$$M = 20 \text{ N} \times 4 \text{ m} = 80 \text{ Nm}$$
Hence, the moment is 80 Nm, illustrating the direct correlation between force, distance from the pivot, and the resulting torque.
Example 2: Using the Principle of Moments
The second example involves a beam supported at two points, where different forces are applied at specified distances. Here, the principle of moments is employed, which states that for a system in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. Given forces of 30 N and 50 N at different locations, students are guided through the process of setting up the equation:
$$30 \times 2 = 50 \times x$$
Solving this equation leads to finding the unknown distance from the left support to the pivot. The calculated distance is:
$$x = \frac{30 \times 2}{50} = 1.2 \text{ m}$$
Thus, the system is balanced if the pivot is placed 1.2 meters from the left support.
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Example 1: Moment of a Force
Chapter 1 of 2
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Chapter Content
A force of 20 N is applied at a distance of 4 meters from the pivot. Find the moment.
M=F×d=20 N×4 m=80 Nm
Hence, the moment is 80 Nm.
Detailed Explanation
In this example, we are tasked with calculating the moment produced by a force. The moment is calculated by using the formula: M = F × d, where M is the moment, F is the force applied, and d is the distance from the pivot point. Here, a force of 20 Newtons is applied at a distance of 4 meters. Thus, substituting the values into the formula gives us M = 20 N × 4 m = 80 Nm. Therefore, the moment is 80 Newton-meters.
Examples & Analogies
Imagine you are trying to open a heavy door. The more you push (the force) and the further away from the hinges (the pivot) that you apply force, the easier the door swings open. In this case, pushing with a force of 20 N at 4 meters away creates a strong turning effect, or moment, making it easier to open the door.
Example 2: Using the Principle of Moments
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Chapter Content
A beam is supported at two points. A force of 30 N is applied 2 meters from the left support, and a force of 50 N is applied 3 meters from the left support. Find the distance from the left support to the pivot where the system is balanced.
Using the principle of moments:
30×2=50×x
x=30×250=60
x=1.2 m
Hence, the pivot point should be placed 1.2 meters from the left support.
Detailed Explanation
In this example, we apply the principle of moments to determine the position of the pivot so that the system remains balanced. We know that for a system in equilibrium, the clockwise moments must equal the anticlockwise moments. Here, a force of 30 N is applied at a distance of 2 m from the left support, creating a moment of 30 N × 2 m = 60 Nm in the clockwise direction. A force of 50 N is applied 3 m from the left support, creating an unknown moment M = 50 N × x (where x is the distance from the left support to the pivot). Setting these moments equal gives us: 30 × 2 = 50 × x. Solving this leads to x = 1.2 m, showing that the pivot point must be placed 1.2 meters from the left support.
Examples & Analogies
Think of this example as a seesaw where two children are sitting at different distances from the pivot. If one child is heavier or sitting further from the center, you need to balance it out by adjusting the distance of the other child or changing their weight. Here, by calculating the distance required from one side to the pivot, we ensure that the seesaw stays level and doesn't tip over.
Key Concepts
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Moment: The rotational effect of a force at a distance from a pivot point.
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Principle of Moments: The condition where clockwise moments equal anticlockwise moments in equilibrium.
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Moment Arm: The distance from the pivot to the line of action of the force.
Examples & Applications
Calculating the moment caused by a 15 N force at a distance of 3 m: M = 15 N × 3 m = 45 Nm.
Determining the unknown pivot location using the applied forces of 20 N and 40 N at different distances.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For a moment that spins, use F and d, at the pivot it begins, M is calculated, easy as can be.
Stories
Imagine a seesaw, with kids at each end, the one sitting far away has more power to bend!
Memory Tools
For remembering the moment formula, think of 'Funny Dogs', for F and d.
Acronyms
MFD
Moment = Force × Distance.
Flash Cards
Glossary
- Moment
The turning effect produced by a force applied at a distance from a pivot point or axis of rotation.
- Torque
Another term for moment, indicating the rotational effect of a force.
- Pivot
The fixed point around which a lever rotates.
- Equilibrium
A state where all forces acting upon an object are balanced, causing no motion.
Reference links
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