Question

Identify the force present and explain whether work is being performed in the following cases: (a) You lift a pencil off the top of a desk. \((\mathbf{b})\) A spring is compressed to half its normal length.

Short answer

In both cases, work is being performed. In Case 1, the force involved is the applied force used to lift the pencil, working against gravitational force. The work done in this case is given by \(W = Fd\), where F is the applied force, and d is the distance the pencil travels upwards. In Case 2, the force involved is the spring force, which acts to restore the spring to its uncompressed state according to Hooke's Law (\(Fs = -kx\)). The work done in compressing the spring is given by \(W = Fsx\), where Fs is the spring force, and x is the compression of the spring.

Step-by-step solution

(Case 1: Lifting a pencil)

In this case, we have two main forces acting on the pencil: gravitational force (mg) and the applied force (F) used to lift the pencil. The gravitational force acts downward, while the applied force acts upward. Since the pencil is being lifted upward, the applied force is greater than the gravitational force. To determine whether work is being done, let's consider the work formula: \[ W = Fd\cos{\theta} \] where W is work, F is the applied force, d is the distance the pencil travels upwards, and θ is the angle between the applied force and the distance. In this case, the applied force and the distance moved by the pencil are in the same direction, so the angle θ is 0 degrees, and cos(0) = 1. Therefore, the work done in this case is: \[ W = Fd \] Since the pencil is being lifted (Fd > 0), work is performed on the pencil in this case.

(Case 2: Compressing a spring)

In this scenario, we have a spring compressed to half of its normal length. The main force involved is the spring force, which acts to restore the spring to its uncompressed state. The spring force is given by Hooke's Law: \[ Fs = -kx \] where Fs is the spring force, k is the spring constant, and x is the compression of the spring. Again, to determine whether work is being done, let's consider the work formula: \[ W = Fd\cos{\theta} \] In this case, the force (Fs) and the distance (x) are in the same direction as the spring is compressed horizontally, so the angle θ is 0 degrees and cos(0) = 1. Therefore, the work done in this case is: \[ W = Fsx \] Since the spring is being compressed (Fsx > 0), work is being performed on the spring in this case.

Key concepts

Gravitational Force

When you lift an object, like a pencil, off a desk, you encounter gravitational force. This force is the weight of the pencil and it acts downward, toward the Earth. In simple terms, gravitational force is the pull that the Earth exerts on any object. It is calculated as the mass of the object multiplied by the acceleration due to gravity: \( F_g = mg \). Here, \( m \) represents the object's mass, while \( g \) is the constant acceleration due to gravity, roughly \( 9.8 \, \text{m/s}^2 \) on Earth's surface.
Gravitational force is always directed towards the center of the Earth. No matter which way you move the pencil, Earth's gravity is pulling it down. But when you successfully lift the pencil, it means you've exerted an opposite force. This opposite force must be greater than or equal to gravitational force for the pencil to move upward.

Applied Force

Applied force is the force you exert when you lift, push, or pull an object. When you lift a pencil, the applied force is in the upward direction. It opposes gravitational force, which pulls the pencil down. To lift the pencil, your applied force must be greater than the gravitational force.
Let's look at how work is involved. Work is calculated with the formula \( W = F \cdot d \cdot \cos(\theta) \). Here, \( F \) is the force you apply, \( d \) is the distance the pencil is lifted, and \( \theta \) is the angle between the force direction and movement direction. In your case, \( \theta \) is \( 0 \) degrees because the force and movement are in the same upward direction.
If there's positive work, like when \( F \cdot d \) is greater than zero, it indicates that energy is being transferred to the pencil, causing it to move.

Spring Force

A spring force is a force that attempts to return a spring to its original length, whether it has been compressed or stretched. This can be illustrated by compressing a spring to half its normal length. The force excerted by the spring, known as the spring force, behaves according to Hooke's Law.
This spring force can be mathematically described with Hooke's Law, using \( F_s = -kx \), where \( F_s \) represents the spring force, \( k \) is the spring constant which is unique to each spring, and \( x \) is the displacement (compression or extension) from the spring's normal length. The negative sign shows that the force is restoring, meaning it seeks to return the spring to its original length.

Hooke's Law

Hooke's Law is fundamental in understanding how springs work. In simple terms, it states that the force exerted by a spring is directly proportional to its extension or compression, up to a certain limit. This is written as \( F_s = -kx \).
Here, the spring constant \( k \) quantifies the stiffness of a spring. A larger \( k \) means a stiffer spring. The compression or extension \( x \) is how far the spring is compressed or stretched from its equilibrium position.
When you apply a force to compress a spring, you are doing work against the spring force. This work is calculated in the same way as before: \( W = F \cdot d \cdot \cos(\theta) \). For a spring, the force and movement are in the same direction (\( \theta = 0 \)), simplifying the calculation to \( W = F_s \cdot x \). The amount of work done is stored as potential energy in the spring.