Problem 2 - Applied Mathematics

Question

A particle of mass \(3 \mathrm{~kg}\) slides down a smooth plane inclined at arcsin \(\frac{1}{3}\) to the horizontal. The acceleration of the particle is: (a) \(\frac{R}{3} \mathrm{~ms}^{-2}\) (b) \(g \mathrm{~ms}^{-2}\) (c) \(1 \mathrm{~ms}^{-2}\) (d) \(3 g \mathrm{~ms}^{-2}\) (e) 0 .

Step-by-Step Solution

Answer

The acceleration of the particle is \(g \text{ ms}^{-2}\).

Step 1 - Identify the Given Values

The mass of the particle is given as \(3 \text{ kg}\). The incline angle of the plane is given as \(\arcsin\left(\frac{1}{3}\right)\).

Step 2 - Calculate the Angle

Given \(\arcsin\left(\frac{1}{3}\right)\), let \(\theta\) be the angle of the incline. This means \(\sin \theta = \frac{1}{3}\).

Step 3 - Identify Forces Acting on the Particle

The force due to gravity acting down the incline is \(mg\sin \theta\), where \(m\) is the mass and \(g\) is the acceleration due to gravity.

Step 4 - Calculate the Component of Gravity

Substitute the values: \(m = 3 \text{ kg}\) and \(\sin \theta = \frac{1}{3}\). Thus, the component of the gravitational force acting down the incline is: \[3g \cdot \frac{1}{3} = g\]

Step 5 - Calculate the Acceleration

As the plane is smooth (frictionless), the only force causing acceleration is the component of gravity down the incline. Therefore, the acceleration \(a\) is \(g\).

Key Concepts

Particle Dynamics

Particle dynamics is the study of forces and their impact on the motion of particles. In this problem, the particle is a small mass sliding down an inclined plane. When analyzing such systems, it's crucial to understand the forces at play and how they affect the particle's motion. Here, the primary force to consider is gravity, which influences how the particle moves down the incline. As the particle slides down the frictionless plane, only the component of gravitational force parallel to the incline affects its acceleration. This setup simplifies our calculations, making it easier to predict the particle's acceleration based on basic principles of dynamics.

Gravitational Force Components

To fully understand the motion of the particle, we need to break down the gravitational force into components. Gravity always acts downwards, towards the center of the Earth. However, because the plane is inclined, we need to consider how this force can be split into two parts:

• The component perpendicular to the plane, which doesn't affect the particle's motion because the plane supports the particle in this direction.
• The component parallel to the plane, which causes the particle to accelerate down the slope.

In the exercise, we use the angle of inclination \theta, given as \theta = \text{arcsin}(\frac{1}{3}). This helps us find the parallel component of the force:
\[ F_{parallel} = mg \text{sin} \theta \] By substituting the given values:
\[ F_{parallel} = 3 \text{kg} \times g \times \frac{1}{3} = g \] Therefore, the component of gravity acting along the incline exactly equals the acceleration due to gravity, g, leading us to the next concept.

Trigonometric Functions in Physics

Trigonometric functions like sine, cosine, and tangent are vital in physics, especially when dealing with inclines or angles. They help us decompose forces and understand how different components contribute to motion. In our problem, the angle of inclination determines how we split the gravitational force into components. Using \theta = \text{arcsin}(\frac{1}{3}), we note that:
\[ \text{sin}(\theta) = \frac{1}{3} \] This allows us to resolve the force of gravity acting down the incline. Specifically, we get:
\[ mg \times \text{sin} \theta = mg \times \frac{1}{3} \] Whenever you deal with inclined planes or any scenario involving angles, remember to use the appropriate trigonometric function. It simplifies the problem by breaking down forces into manageable components, ultimately leading to a clearer understanding of dynamics and motion. This technique is widely applicable in countless physics problems involving forces and motion on inclined surfaces.