Problem 14 - Applied Mathematics

Question

(a) A particle is moving with a constant velocity. (b) The forces acting on a particle are in equilibrium.

Step-by-Step Solution

Answer

A particle with constant velocity has zero net force. Forces in equilibrium mean the net force is zero.

Step 1: Understand Constant Velocity

A particle moving with a constant velocity means that its speed and direction do not change over time. This implies there is no acceleration.

Step 2: Apply Newton's First Law

Newton's First Law states that an object in motion will stay in motion with a constant velocity if the net force acting on it is zero. Thus, if the particle is moving with a constant velocity, the total (net) force acting on it must be zero.

Step 3: Define Equilibrium in Forces

When forces acting on a particle are in equilibrium, it means the sum of all forces is zero. Mathematically, this is represented as \( \boldsymbol{F}_{\text{net}} = \boldsymbol{0} \) where \( \boldsymbol{F}_{\text{net}} \) is the vector sum of all forces.

Step 4: Equilibrium Condition

Since the forces are in equilibrium (part b), the net force acting on the particle is zero (\( \boldsymbol{F}_{\text{net}} = \boldsymbol{0} \)). This condition is also consistent with part (a) because a particle moving with a constant velocity has no net force acting on it.

Key Concepts

constant velocity

When we talk about constant velocity, we mean that an object's speed and direction remain unchanged over time. Imagine driving a car at 60 mph straight on an empty highway without any acceleration or deceleration. That car is moving at a constant velocity.

Newton's First Law plays a crucial role here. It states that an object will continue in its state of rest or uniform motion (constant velocity) unless acted upon by a net external force. In simpler terms, if no overall force is pushing or pulling the object, it will neither speed up nor slow down.

For a particle to maintain a constant velocity, there must be zero net force acting on it. This means all forces acting on the particle balance each other out perfectly. So, if you see a particle moving at a constant velocity, you can be sure the net forces are in equilibrium.

equilibrium of forces

Equilibrium of forces occurs when all the forces acting on a particle cancel each other out, resulting in a net force of zero. Think of it like a tug-of-war where both sides pull with equal force, causing no movement in either direction.

Mathematically, this condition is represented as \(\boldsymbol{F}_{\text{net}} = \boldsymbol{0} \), where \(\boldsymbol{F}_{\text{net}} \) is the vector sum of all the forces.

When forces are in equilibrium, two important things happen:

• The object either remains at rest if it was initially at rest.
• It continues to move at a constant velocity if it was already in motion.

Understanding equilibrium helps in solving many physical problems, from the motion of cars to the stability of buildings and bridges.

net force

Net force is a fundamental concept in physics that describes the overall force acting on an object. It is the sum of all individual forces applied to the object, taking both magnitude and direction into account.

Newton's First Law tells us that an object will stay at rest or move with a constant velocity unless acted upon by a net force. So, if the net force is zero, the object will not accelerate.

To find the net force, you need to:

• Identify all individual forces acting on the object.
• Add the forces vectorially, considering both magnitude and directions.

For example, if two forces, one of 5 N to the right and another of 5 N to the left, act on a particle, the net force is 0 N. This means the particle will either stay at rest or keep moving with a constant velocity if it was already in motion.