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. (b) A spring is compressed to half its normal length.

Short answer

In both cases, work is being performed. (a) In lifting a pencil off the desk, the force exerted by your hand opposes gravity, and the pencil is displaced vertically upwards. (b) In compressing a spring, an external force acts inwards, displacing the spring from its normal length to half its normal length.

Step-by-step solution

Case (a): Lifting a pencil off the top of a desk

1. Identify the forces acting on the pencil: There are two forces acting on the pencil - the gravitational force acting downwards and the force exerted by your hand acting upwards (opposite to the gravitational force). 2. Find if there is any displacement: When you lift the pencil off the desk, it is being displaced vertically upwards. 3. Check if work is being performed: Yes, work is being performed in this case because there is a force acting on the pencil (the force exerted by your hand) and the pencil is being displaced in the same direction as the force (vertically upwards).

Case (b): A spring compressed to half its normal length

1. Identify the forces acting on the spring: When the spring gets compressed it means an external force is being applied inwards. The force from the spring is also acting in the opposite direction (outwards) due to Hooke's Law. 2. Find if there is any displacement: The spring is being displaced from its normal length to half its normal length, meaning that there is a displacement. 3. Check if work is being performed: Yes, work is being performed in this case because there is an external force acting on the spring (compressing it) and the spring is being displaced in the same direction as the force (inwards).

Key concepts

Forces

In physics, a force is any interaction that, when unopposed, changes the motion of an object. Forces can be divided into contact and non-contact forces. Contact forces require physical interaction, like the push of a hand on a pencil or the squeeze of a spring. Non-contact forces include gravity pulling objects toward the Earth.

For example, when you lift a pencil off a desk, two primary forces act on it:

• Gravitational force - pulling the pencil downward.
• Exerted force by the hand - pushing the pencil upward.

These forces are vector quantities. They have both magnitude and direction. In our case, the force exerted by the hand must be greater than the gravitational force for the pencil to move up, showcasing an instance where force causes motion, illustrating the work done by the force.

Displacement

Displacement refers to an object's change in position. It is a vector quantity, possessing both magnitude and direction. This concept is vital to understanding if work is performed, as work results from force causing displacement in the same direction.

Consider the scenario of lifting a pencil. The pencil's path, moving vertically upward from the desk, embodies displacement. Movement in the same direction of the applied force is one criterion for work to occur.

Similarly, when a spring is compressed, displacement occurs inwards from its original position. This inward movement aligns with the external force applied, leading to the conclusion that work is conducted. Displacement acts as the cornerstone in determining the presence of work alongside force.

Hooke's Law

Hooke's Law is a principle describing the behavior of springs. It states that the force needed to compress or extend a spring is proportional to the displacement from its original length. Mathematically, it is expressed as:\[ F = -kx \]where:

• \( F \) is the force applied to the spring,
• \( k \) is the spring constant (describes the stiffness of the spring),
• \( x \) is the displacement of the spring from its equilibrium position.

In the scenario where a spring is compressed to half its normal length, an external force compresses the spring. The spring, according to Hooke's Law, exerts an equal and opposite force. This force is known as the restoring force and acts outwards, attempting to return the spring to its natural length. Understanding Hooke's Law is crucial to comprehend how springs behave under force, emphasizing the relationship between displacement and force, both fundamental to executing and detecting work.