Problem 9 - Applied Physics

Question

A worker lifts 75 concrete blocks a distance of \(1.50 \mathrm{~m}\) to the bed of a truck. Each block has a mass of \(4.00 \mathrm{~kg} .\) How much work is done to lift all the blocks to the truck bed?

Step-by-Step Solution

Answer

The total work done is 4410 J.

Step 1: Understanding the Problem

We need to calculate the work done by the worker in lifting the concrete blocks to the truck bed. The work is given by the formula \( W = F \cdot d \), where \( F \) is the force and \( d \) is the distance. Here, the force is equal to the gravitational force acting on the blocks.

Step 2: Calculate the Force for One Block

Each block has a mass \( m = 4.00 \mathrm{~kg} \). The force needed to lift one block is the gravitational force, which is calculated as \( F = m \cdot g \), where \( g = 9.8 \mathrm{~m/s^2} \) (acceleration due to gravity). Thus, \( F = 4.00 \cdot 9.8 \).

Step 3: Calculate Work for One Block

The work done to lift one block is \( W = F \cdot d \). Substituting the values \( F = 4.00 \cdot 9.8 \) and \( d = 1.50 \mathrm{~m} \), the work done for one block is \( W = (4.00 \cdot 9.8) \cdot 1.5 \).

Step 4: Calculate Total Work for All Blocks

Since there are 75 blocks, the total work done is 75 times the work done for one block. If \( W_{ ext{one block}} = 4.00 \cdot 9.8 \cdot 1.5 \), then \( W_{ ext{total}} = 75 \cdot W_{ ext{one block}} \).

Step 5: Final Calculation

Calculate the numerical value: 1. Work for one block: \( W_{ ext{one block}} = 4.00 \cdot 9.8 \cdot 1.5 = 58.8 \mathrm{~J} \). 2. Total work: \( W_{ ext{total}} = 75 \cdot 58.8 = 4410 \mathrm{~J} \).

Key Concepts

Gravitational Force

Gravitational force is the force that attracts two bodies toward each other, but we commonly associate it with the force that pulls objects toward the Earth's surface. This force gives weight to physical objects and explains why they fall when dropped.
For the concrete blocks in the exercise, the gravitational force is crucial in understanding how much work is needed to lift them. The force acting on each block is the product of its mass and the acceleration due to gravity. This relationship is expressed by the formula:

• \( F = m \cdot g \)

where \( F \) is the force, \( m \) is the mass of the block, and \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth).
In the context of lifting the blocks, the gravitational force must be overcome to do work, moving them against Earth’s pull.

Mass and Weight

People often confuse mass and weight, but they are distinct concepts. Mass is the amount of matter in an object, and it remains constant regardless of location. It's measured in kilograms (kg) and is a fundamental property of the object.
Weight, on the other hand, is the force exerted by the gravitational pull on that mass. Since it's a force, weight depends on the gravitational field strength where the object is. The weight \( W \) of an object can be calculated using:

• \( W = m \cdot g \)

In our example, the block's mass of \( 4.00 \text{ kg} \) becomes heavier where the gravitational pull is stronger, but on Earth, the weight is simply the mass multiplied by Earth’s gravitational acceleration.
Understanding this helps in calculating the work to lift the concrete blocks since you need to know the force applied, which indeed is their weight under gravitational pull.

Distance and Displacement

Distance and displacement are both measures of movement, but they are not identical. Distance is the total path taken by an object, regardless of direction. It’s a scalar quantity because it only has magnitude.

• Example: Walking around a circular track involves covering a distance, even if you end where you started.

Displacement, on the other hand, is a vector quantity that considers only the change in position from the starting point to the endpoint, including direction.
In the exercise, the worker lifts the blocks by a vertical displacement of \(1.50 \text{ m}\). This movement is directly upward—a straight line path is the same as the total distance.

• Thus, the displacement is equal to the distance in this context because there is no change in direction.

This value is crucial in the work formula \( W = F \cdot d \), where \( d \) represents displacement, ensuring accurate work calculation since the force must move an object over a specific displacement.