Problem 16 - Applied Physics

Question

A bar is used to lift a \(10 \overline{0}\) -kg block of concrete. The pivot is \(1.00 \mathrm{~m}\) from the block. (a) If the worker pushes down on the other end of the bar a distance of \(2.50 \mathrm{~m}\) from the pivot, what force (in \(\mathrm{N}\) ) must the worker apply? (b) Find the MA.

Step-by-Step Solution

Answer

The worker must apply 3920 N, and the MA is 2.5.

Step 1: Understanding the Problem

We are tasked with finding the force a worker needs to apply on a bar to lift a block weighing 1000 kg using a lever. The pivot point is 1.00 m from the block, and the worker applies the force at a distance of 2.50 m from the pivot. We will also find the mechanical advantage (MA) of this arrangement.

Step 2: Identifying the Forces

Firstly, identify the weight of the block. The weight, which acts as a force, is given by the formula: \( F = mg \), where \( m = 1000 \) kg and \( g = 9.8 \text{ m/s}^2 \). This gives \( F = 1000 \times 9.8 \). The lever arm distance for the block is 1.00 m.

Step 3: Calculating the Torque Due to the Block

Torque is given by the product of force and lever arm: \( \tau = F \times d \). For the block, \( \tau = 1000 \times 9.8 \times 1.00 \). Calculate to determine the torque exerted by the block on the pivot.

Step 4: Setting Up the Torque Balance Equation

At equilibrium, the clockwise torque by the worker must balance the counterclockwise torque by the block. Therefore, \( F_{worker} \times 2.50 = 1000 \times 9.8 \times 1.00 \). Solve for \( F_{worker} \).

Step 5: Solving for the Force Applied by the Worker

Rearrange the torque balance equation to find \( F_{worker} \): \( F_{worker} = \frac{1000 \times 9.8 \times 1.00}{2.50} \) N. Calculate \( F_{worker} \).

Step 6: Finding the Mechanical Advantage (MA)

The mechanical advantage of a lever is given by the ratio of the force exerted by the load (weight of the block) to the force exerted by the worker: \( MA = \frac{F_{block}}{F_{worker}} \). Use the known values to calculate the MA.

Key Concepts

Torque Calculations

Torque is a pivotal concept in lever mechanics. It refers to the measure of the force that causes an object to rotate about an axis. In this exercise, the axis is the pivot point of the lever. Torque is calculated using the formula: \( \tau = F \times d \), where \( \tau \) is the torque, \( F \) is the force applied, and \( d \) is the distance from the pivot.

For the block in our exercise, the torque is generated by its weight: a force due to gravity. Since the block weighs 1000 kg, and the gravitational acceleration \( g \) is 9.8 m/s², the force acting downward is \( F = 1000 \times 9.8 \) Newtons. Given the length from the pivot to the block is 1.00 m, the block's torque is \( 1000 \times 9.8 \times 1.00 \). This determines how strongly the block pulls on its arm of the lever.

The worker, on the other hand, needs to apply a force to generate an equivalent, but opposite, torque to lift the block. They apply this force at a 2.50 m distance from the pivot. To find the needed force \( F_{worker}\), one would use the torque balance equation, ensuring the overall system reaches equilibrium.

Mechanical Advantage

Mechanical advantage (MA) is a crucial element in understanding how levers make lifting or moving heavy objects easier. Essential in evaluating the effectiveness of a lever, mechanical advantage is defined as the ratio of the output force to the input force.

In simple terms, it tells us how much a lever amplifies the input force. The formula is: \( MA = \frac{F_{output}}{F_{input}} \). Here, \( F_{output} \) is the force exerted by the block (or weight), and \( F_{input} \) is the force applied by the worker.

For our exercise, the \( F_{output} \) is simply the weight of the block: \( 1000 \times 9.8 \) Newtons. The force \( F_{input} \) that the worker needs to apply can be calculated using the torque balance equation. Once you have both forces, you can compute the mechanical advantage, which tells you how easy the lever makes the lifting task.

Force Equilibrium

Force equilibrium is quite significant in lever mechanics when balancing opposing forces. In the context of this exercise, force equilibrium means the sum of all torques acting around a pivot is zero. This indicates that the block doesn't move and the system is stable.

The block applies a counterclockwise torque due to its weight, while the worker's task is to apply a force that creates an equal and opposite clockwise torque. Consider the equation: \( F_{worker} \times 2.50 = 1000 \times 9.8 \times 1.00 \).

This equilibrium condition ensures that when the worker applies the calculated force, the torques balance each other out, allowing the block to lift smoothly without sudden shifts.

• The counterclockwise torque generated by the block: \( 1000 \times 9.8 \times 1.00 \)
• The clockwise torque needed from the worker: \( F_{worker} \times 2.50 \)

Solving this scenario uses the principle of leveraging different arm distances for force multiplication and ease of lifting.