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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

Expert verified
The student exerts a force of approximately 33.68 N.

Step by step solution

01

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 \(W = F \times d\).
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

You push a book \(0.45 \mathrm{~m}\) across a desk with a \(5.2-\mathrm{N}\) force that is at an angle of \(21^{\circ}\) below the horizontal. How much work did you do on the book?

Cookie Power To make a batch of cookies, you mix half a bag of chocolate chips into a bowl of cookie dough, exerting a \(21-\mathrm{N}\) force on the stirring spoon. Assume that your force is always in the direction of motion of the spoon. (a) What power is needed to move the spoon at a speed of \(0.23 \mathrm{~m} / \mathrm{s}\) ? (b) How much work do you do if you stir the mixture for \(1.5\) min?

Think \& Calculate A grandfather clock is powered by the descent of a \(4.35-\mathrm{kg}\) weight. (a) If the weight descends through a distance of \(0.760 \mathrm{~m}\) in \(3.25\) days, how much power does it deliver to the clock? (b) To increase the power delivered to the clock, should the time it takes for the mass to descend be increased or decreased? Explain.

Calculate A \(9.50-\mathrm{g}\) bullet has a speed of \(1.30 \mathrm{~km} / \mathrm{s}\). What is the kinetic energy of the bullet?

Think \& Calculate A pitcher accelerates a 0.14-kg hardball from rest to \(25.5 \mathrm{~m} / \mathrm{s}\) in \(0.075 \mathrm{~s}\). (a) How much work does the pitcher do on the ball? (b) What is the pitcher's power output during the pitch? (c) Suppose the ball reaches \(25.5 \mathrm{~m} / \mathrm{s}\) in less than \(0.075 \mathrm{~s}\). Is the power produced by the pitcher in this case more than, less than, or the same as the power found in part (b)? Explain.

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