Question

To move a suitcase up to the check-in stand at an airport, a student pushes with a horizontal force through a distance of \(0.95 \mathrm{~m}\). If the work done by the student is \(32 \mathrm{~J}\), what is the magnitude of the force he exerts?

Short answer

The student exerts a force of approximately 33.68 N.

Step-by-step solution

Identify the formula for work done

The general formula for work done is given by: \[ W = F imes d imes \cos(\theta) \]where \(W\) is the work done, \(F\) is the force applied, \(d\) is the distance moved, and \(\theta\) is the angle between the force and the direction of movement. Since the force is horizontal, \(\theta = 0\) and \(\cos(0) = 1\), simplifying the formula to \(W = F \times d\).

Substitute the given values into the formula

Substitute the known values into the simplified formula: \[ 32 \mathrm{~J} = F \times 0.95 \mathrm{~m} \] We have \(W = 32\) Joules and \(d = 0.95\) meters.

Solve for the force \(F\)

Now, solve for \(F\) by rearranging the equation: \[ F = \frac{W}{d} \] Substitute the known values:\[ F = \frac{32 \mathrm{~J}}{0.95 \mathrm{~m}} \approx 33.68 \mathrm{~N} \] Thus, the magnitude of the force exerted by the student is approximately \(33.68\) Newtons.

Key concepts

Force

Force is a fundamental concept in physics. It represents a push or pull on an object, creating a change in the object's motion or state of rest. In the scenario of moving a suitcase, the student applies a specific force to overcome the inertia of the suitcase and move it towards the check-in stand.

Force is usually measured in Newtons (N). One Newton is defined as the force necessary to accelerate a one-kilogram mass by one meter per second squared. It is often calculated using the equation:

• \[ F = ma \]

where \( m \) is the mass of the object and \( a \) is the acceleration.

However, in the given exercise, the force refers to the horizontal push necessary to perform work on the suitcase, which involves moving it a certain distance. The following formula relates force to work done:

• \[ W = F \times d \times \cos(\theta) \]

This specific calculation in our exercise simplifies since the angle is zero, meaning the force is applied directly in the direction of movement.

Distance

The distance in the context of the work done is the measure of how far the force has moved the object in the direction of the force. It is one of the crucial factors in determining the work done on an object.

Distance is linearly proportional to work, meaning that the farther you move an object while exerting force, the more work you do. In this case, the student moves the suitcase a distance of 0.95 meters. This distance directly impacts the total work done, as shown in the formula:

• \[ W = F \times d \times \cos(\theta) \]

For the suitcase example, as soon as distance is known, it can be substituted into the work equation after accounting for the angle \( \theta \). The calculation becomes straightforward since any work calculation requires knowing how far an object has traveled under the influence of a force.

Angle Between Force and Direction

The angle between the force and the direction of movement significantly affects the calculation of work done. In physics, this is accounted for using the cosine of the angle (\( \theta \)) in the work done formula:

• \[ W = F \times d \times \cos(\theta) \]

When the angle is 0°, it indicates that the force is entirely in the direction of movement. This situation maximizes the work done because the cosine of 0° is 1.

In other scenarios where the force is not aligned with the direction of movement, the angle \( \theta \) reflects how much of that force contributes to moving the object. The more the angle deviates from 0°, the less effective the force becomes in producing work, since \( \cos(\theta) \) becomes less than 1.

For this particular exercise, given that the force is horizontal and the suitcase moves horizontally, \( \theta = 0 \), which simplifies the work equation to indicate maximum efficiency of the force in the direction of movement.