Problem 7 - Applied Mathematics

Question

A particle is moving with uniform velocity. (a) The forces acting on the particle are in equilibrium. (b) The particle has zero acceleration. (c) There is a resultant force acting on the body in the direction of the velocity.

Step-by-Step Solution

Answer

Statements (a) and (b) are true. Statement (c) is false.

Step 1: Understanding Uniform Velocity

When a particle is moving with uniform velocity, it means that the particle is moving in a straight line with constant speed. There is no change in its speed or direction.

Step 2: Analyzing Equilibrium of Forces

For a particle to move with uniform velocity, the forces acting on the particle must be balanced. This implies that the net or resultant force acting on the particle is zero. Therefore, statement (a) 'The forces acting on the particle are in equilibrium' is true.

Step 3: Acceleration Analysis

Since the particle is moving with a constant velocity, its acceleration is zero (acceleration is the rate of change of velocity). Therefore, statement (b) 'The particle has zero acceleration' is true.

Step 4: Revisiting Resultant Force

Given that the forces are in equilibrium and the particle’s acceleration is zero, there cannot be a resultant force acting on the particle. Hence, statement (c) 'There is a resultant force acting on the body in the direction of the velocity' is false.

Key Concepts

equilibrium of forces

When we talk about *equilibrium of forces*, it means that all the forces acting on a particle balance each other out. Imagine a tug-of-war game where both teams have equal strength and no one is winning. Here, the forces pulling on the particle from different directions cancel each other, resulting in no net force.
This balance of forces is crucial for a particle to move with uniform velocity. If any unbalanced force was present, it would cause the particle to accelerate or decelerate.
For example, if you're holding a book still in the air, the force of your hand pushing upward balances the weight of the book pulling downward. Thus, the book remains in equilibrium and doesn't move.

zero acceleration

A particle has *zero acceleration* when its velocity remains constant over time. In simpler terms, the speed and direction of the particle don't change. This happens when there is no external force acting on it to alter its motion.
Acceleration is the rate of change of velocity. For instance, touching the gas pedal in a car increases its velocity, which means the car accelerates.
When moving with zero acceleration, the particle's velocity graph would be a straight horizontal line, indicating no change over time. People often get confused and think zero acceleration means the particle is not moving, but it can indeed be moving at a constant velocity.

resultant force

A *resultant force* is the overall force acting on a particle when you combine all the individual forces. For a particle to move with uniform velocity, this resultant force must be zero.
If the resultant force equals zero, it implies all acting forces are balanced, putting the particle in equilibrium, as discussed.
However, if there is any resultant force present, it will cause the particle to accelerate in the direction of that force. This is because, according to Newton's Second Law of Motion, Force equals mass times acceleration \( F = ma \). The resultant force introduces acceleration or deceleration, changing the particle's velocity.
Understanding the resultant force helps explain why a particle continues moving in a straight line at constant speed when no net force acts upon it.