Problem 2 - A History of Mathematics: An Introduction

Question

If a weight of \(8 \mathrm{~kg}\) is placed \(10 \mathrm{~m}\) from the fulcrum of a lever and a weight of \(12 \mathrm{~kg}\) is placed \(8 \mathrm{~m}\) from the fulcrum in the opposite direction, toward which weight will the lever incline?

Step-by-Step Solution

Answer

Answer: The lever will incline towards the 12 kg weight.

Step 1: Calculate the moment of force for the 8 kg weight

To calculate the moment of force for the 8 kg weight, we will multiply its weight by its distance from the fulcrum: \( 8 \mathrm{~kg} \times 10 \mathrm{~m}\). Moment of force \(= 8 \times 10 = 80 \mathrm{~kg \cdot m}\).

Step 2: Calculate the moment of force for the 12 kg weight

To calculate the moment of force for the 12 kg weight, we will multiply its weight by its distance from the fulcrum: \(12 \mathrm{~kg} \times 8 \mathrm{~m}\). Moment of force \(= 12 \times 8 = 96 \mathrm{~kg \cdot m}\).

Step 3: Compare the moments of force

Now, we will compare the moments of force calculated in steps 1 and 2 to determine towards which weight the lever will incline. Moment of force for 8 kg weight \(= 80 \mathrm{~kg \cdot m}\). Moment of force for 12 kg weight \(= 96 \mathrm{~kg \cdot m}\). Since the moment of force for the 12 kg weight is greater than the moment of force for the 8 kg weight (96 > 80), the lever will incline towards the 12 kg weight.

Key Concepts

Levers

A lever is a simple machine that helps us lift or move loads with less effort. Imagine a seesaw; that's a common example of a lever. Levers have three main parts:

• **Fulcrum:** The pivot point around which the lever turns.
• **Effort Arm:** The side where force is applied to lift the load.
• **Load Arm:** The side where the load is placed, opposite to the effort.

The effectiveness of a lever depends on the position of these parts.
By varying the distances of the effort and load arms from the fulcrum, levers can reduce the amount of force needed to lift a heavy load. In the given exercise, the lever tilts towards the heavier force, determining which side of the seesaw goes down.

Moments of force

Moments of force, also known simply as moments, are vital in understanding how levers work.
A moment is the turning effect produced by a force acting at a distance from the pivot point.
The formula to find the moment of force is:\[\text{Moment of Force} = \text{Force} \times \text{Distance from Fulcrum}\]Key points about moments of force include:

• **Direction matters:** Moments can cause clockwise or counterclockwise rotation.
• **Balance is crucial:** For equilibrium (balance), the sum of clockwise moments should equal the sum of counterclockwise moments.

In the exercise, we calculated the moments for both weights to determine how the lever would tilt. The moment for the 12 kg weight was larger, causing the lever to incline towards that side.

Equilibrium

Equilibrium in the context of levers means that the lever is at rest or balanced. This occurs when there are no net forces causing movement around the fulcrum.
To achieve equilibrium, the moments on either side of the fulcrum must be equal. Understanding equilibrium involves:

• **Sum of moments:** The sum of moments on one side must equal the sum on the opposite side for balance.
• **Static condition:** No movement occurs when the system is in equilibrium.
• **Adjustment of positions:** Changing distances from the fulcrum or altering the weights can achieve balance.

In the given exercise, equilibrium would mean both weights produce the same moment, keeping the lever horizontal. However, since the forces were unequal, the lever inclined towards the side with the larger moment.