Problem 4 - Applied Mathematics

Question

A particle of mass \(m\) slides a distance \(d\) down a plane inclined at \(\theta\) to the horizontal. The work done by the normal reaction \(R\) is: (a) \(R d\) (b) \(m g d \cos \theta\) (c) 0 (d) \(m g d \sin \theta\).

Step-by-Step Solution

Answer

The work done by the normal reaction force is 0.

Step 1: Understand the Problem

Identify what is asked in the problem. We need to find the work done by the normal reaction force on a particle sliding down an inclined plane.

Step 2: Identify the Normal Reaction Force

The normal reaction force, denoted as R, acts perpendicular to the inclined plane.

Step 3: Work Done by Normal Force

Recall that work done by a force is given by the dot product of the force and the displacement vectors, which can also be expressed as the force times the component of the displacement in the direction of the force. Since \( \theta\) is the angle between the direction of the normal force and the displacement, the displacement in the direction of the normal force is zero.

Step 4: Conclusion

Since the displacement component in the direction of the normal reaction force is zero, the work done by the normal reaction force must be zero.

Key Concepts

normal reaction force

The normal reaction force, often known as the normal force, is a crucial concept. It is the force exerted by a surface, perpendicular to the object in contact with it.
In the context of an inclined plane, the normal reaction force acts perpendicular to the surface of the plane.
If you imagine a particle placed on an inclined plane, gravity pulls it down vertically while the inclined plane exerts a normal force perpendicular to its surface.
This force ensures the particle does not pass through the plane, balancing the vertical component of gravity.

inclined plane

An inclined plane is a flat surface tilted at an angle to the horizontal. This simple machine allows studying various forces and motions. When a particle or an object is placed on it, several forces act on it: gravitational force, normal reaction force, and frictional force (if friction is present).
Gravity can be resolved into two components on an inclined plane:

• Parallel to the plane: This pulls the object down the slope.
• Perpendicular to the plane: Balanced by the normal reaction force.

Understanding how these components interact helps analyze movements and energy transitions on inclined surfaces.

work and energy

Work and energy are essential concepts in physics. Work is defined as the transfer of energy by a force causing displacement.
Mathematically, work is represented as:

\( W = \textbf{F} \bullet \textbf{d} \)
Here,
Work depends on:

• The magnitude of the force applied.
• The displacement caused by the force.
• The angle between the force and displacement directions.

In the context of an inclined plane, the work done by the normal reaction force on a particle is zero.
This is because the displacement in the direction of the normal force is zero (as outlined in the solution).

dot product

The dot product is a mathematical operation that takes two vectors and returns a scalar.
It is instrumental in calculating work done by forces.
The dot product of two vectors is given by:
\( \textbf{F} \boldsymbol{\bullet} \textbf{d} = |\textbf{F}| |\textbf{d}| \text{cos}(\theta) \)
where:

• \( |\textbf{F}| \) is the magnitude of the force.
• \( |\textbf{d}| \) is the magnitude of the displacement.
• \( \theta \) is the angle between the force and displacement directions.

In the provided exercise, since the normal reaction force is perpendicular to the displacement (\( \theta = 90^\text{o} \)), the cosine of 90 degrees is zero.
Hence, the work done (dot product) by the normal reaction force is zero.