Question

An object with mass $ 500 \, g $ moves along x-axis with speed $ v = 4\sqrt{x} \, m/s $. The force acting on the object is :

• A. 8 N
• B. 5 N
• C. 6 N
• D. 4 N

Hint:

When the velocity is given as a function of position, use the chain rule for differentiation to find the acceleration: \( a = v \frac{dv}{dx} \). Then, apply Newton's second law \( F = ma \) to find the force.

Answer 1

The Correct Option is D

To solve the problem of determining the force acting on the object, we start by identifying the given parameters and applying relevant physics principles.

1. The mass of the object is given as \(500 \, \text{g}\). As standard practice, we convert the mass into kilograms for calculations in the SI unit system:

\(m = 0.5\, \text{kg}\)

1. The speed of the object is described by the equation \(v = 4\sqrt{x}\), where \(x\) is the position along the x-axis.
2. To find the force acting on the object, we need to use Newton's second law of motion, which states:

\(F = m \cdot a\)

1. First, we need to find the acceleration \(a\), which is the derivative of the velocity \(v\) with respect to time \(t\). However, \(v\) is given as a function of \(x\), so we'll use the chain rule to express acceleration in terms of \(x\).
2. First, compute the derivative of \(v\) with respect to \(x\):

\(\frac{dv}{dx} = \frac{d}{dx}(4\sqrt{x}) = \frac{4}{2\sqrt{x}} = \frac{2}{\sqrt{x}}\)

1. Since acceleration \(a\) is the derivative of velocity with respect to time, we express it using the chain rule:

\(a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v = \left(\frac{2}{\sqrt{x}}\right) \cdot (4\sqrt{x}) = 8\)

1. We find that the acceleration \(a = 8 \, \text{m/s}^2\).
2. Now, applying Newton's second law:

\(F = m \cdot a = 0.5 \cdot 8 = 4 \, \text{N}\)

1. Thus, the force acting on the object is \(4 \, \text{N}\), which matches the correct answer option.

Answer 2

Step 1: Convert the mass to SI units.
The mass of the object is \( m = 500 \, g = 0.5 \, kg \). 
Step 2: Find the acceleration of the object.
The speed of the object is given by \( v = 4\sqrt{x} \). 
To find the acceleration \( a \), we use the chain rule: \[ a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \] First, find \( \frac{dv}{dx} \): \[ \frac{dv}{dx} = \frac{d}{dx} (4\sqrt{x}) = 4 \cdot \frac{1}{2\sqrt{x}} = \frac{2}{\sqrt{x}} \] Now, substitute \( v \) and \( \frac{dv}{dx} \) into the expression for acceleration: \[ a = (4\sqrt{x}) \cdot \left(\frac{2}{\sqrt{x}}\right) = 8 \, m/s^2 \] The acceleration of the object is constant and equal to \( 8 \, m/s^2 \) along the x-axis. 
Step 3: Calculate the force acting on the object using Newton's second law.
According to Newton's second law of motion, the force \( F \) acting on an object is equal to the product of its mass \( m \) and its acceleration \( a \): \[ F = m \cdot a \] Substituting the values of mass and acceleration: \[ F = (0.5 \, kg) \cdot (8 \, m/s^2) = 4 \, kg \cdot m/s^2 = 4 \, N \] The force acting on the object is \( 4 \, N \).