Question

A \(95-\mathrm{kg}\) refrigerator rests on the floor. How much work is required to move it at constant speed for \(4.0 \mathrm{~m}\) along the floor against a friction force of \(180 \mathrm{~N} ?\)

Short answer

Answer: 720 J

Step-by-step solution

Identify the given values

Here we are given three quantities: 1. Mass of the refrigerator, \(m = 95 \thinspace kg\) 2. Distance to be moved, \(d = 4.0 \thinspace m\) 3. Friction force, \(F_{friction} = 180 \thinspace N\)

Apply the work-energy theorem

The work-energy theorem states that work done is equal to the change in kinetic energy. For a given force acting over a distance, the work done can be calculated by \(W = F \cdot d \cdot \cos \theta\), where \(\theta\) is the angle between the force and the direction of movement. Since we are moving the refrigerator against the friction force and they are in opposite directions, the angle between the force vector and the direction of the movement is \(180^{\circ}\). Thecosine of this angle is -1, which implies that \(\cos \theta = -1\).

Calculate the work against friction force

Now, we can calculate the work done against the friction force using the formula mentioned above: \(W = F_{friction} \cdot d \cdot \cos \theta\) Substituting the given values into the equation, we get: \(W = 180 \thinspace N \cdot 4.0 \thinspace m \cdot (-1)\)

Solve for work done

Now, we just need to multiply these values to find the work done: \(W = -720 \thinspace J\) The negative sign indicates that work is done against the friction force. Therefore, \(720 \thinspace J\) of work is required to move the refrigerator at constant speed for \(4.0 \thinspace m\) along the floor against a friction force of \(180 \thinspace N\).

Key concepts

Work Done Calculation

When we talk about the work done calculation, we refer to computing the amount of energy transferred by a force moving an object over a distance. In physics, this concept is crucial because it relates directly to energy transfer.

For example, if you need to move a heavy object, such as a refrigerator, and you know the force required as well as the distance you need to move it, you can calculate the work done using the formula: \[ W = F \times d \times \text{cos}(\theta) \] where W is the work done, F is the force applied, d is the distance moved by the object, and \theta is the angle between the force and the direction of movement.

In our example with the refrigerator, the force of friction opposes the movement, and the angle \( \theta \) is \(180^\text{°}\), making \( \text{cos}(\theta) = -1 \) due to the force's direction being opposite to the movement. The work done is thus calculated as a force of \( 180\thinspace N \) multiplied by a distance of \( 4.0\thinspace m \) and the cosine of \( 180^\text{°} \) which is a simple multiplication but conveys a deep concept regarding energy transfer.

Force and Movement

The interplay between force and movement is at the heart of many physics problems, including how work is done. Force is essentially a push or a pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of them. The effect of this force can cause an object to accelerate, decelerate, remain stationary, or change its direction of movement.

A constant speed indicates that the force applied to move the refrigerator is balanced by the friction force resisting the movement, leading to no net acceleration. Importantly, the magnitude of the friction force tells us how much force needs to be applied to keep the refrigerator moving at a constant speed. This is a foundational concept in understanding dynamics and how forces result in motion or maintain a state of motion.

Kinetic Energy Change

The concept of kinetic energy change is intertwined with work done since work involves transferring energy to an object by force, which can result in an increase or decrease in the object’s kinetic energy. Kinetic energy is the energy an object possesses due to its motion.

According to the work-energy theorem, the work done on an object is equal to the change in kinetic energy. If an object starts at rest and gains speed, its kinetic energy increases, which corresponds to positive work being done on it. Conversely, if the object slows down, work is done against the object's motion, indicated by a negative work value.

In our textbook problem, since the refrigerator is moving at constant speed, the kinetic energy does not change; the work done goes into overcoming the dissipative force of friction. Hence, while there is no change in kinetic energy, work is still being done to maintain the constant speed against the friction. Understanding that work, force, and kinetic energy are fundamentally linked is essential for comprehending energy conservation and transformation processes.