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 :
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- 8 N
- 5 N
- 6 N
- 4 N
The Correct Option is D
Approach Solution - 1
To solve the problem of determining the force acting on the object, we start by identifying the given parameters and applying relevant physics principles.
- 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:
\(m = 0.5\, \text{kg}\)
- The speed of the object is described by the equation \(v = 4\sqrt{x}\), where \(x\) is the position along the x-axis.
- To find the force acting on the object, we need to use Newton's second law of motion, which states:
\(F = m \cdot a\)
- 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 \(x\).
- First, compute the derivative of \(v\) with respect to \(x\):
\(\frac{dv}{dx} = \frac{d}{dx}(4\sqrt{x}) = \frac{4}{2\sqrt{x}} = \frac{2}{\sqrt{x}}\)
- Since acceleration \(a\) is the derivative of velocity with respect to time, we express it using the chain rule:
\(a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v = \left(\frac{2}{\sqrt{x}}\right) \cdot (4\sqrt{x}) = 8\)
- We find that the acceleration \(a = 8 \, \text{m/s}^2\).
- Now, applying Newton's second law:
\(F = m \cdot a = 0.5 \cdot 8 = 4 \, \text{N}\)
- Thus, the force acting on the object is \(4 \, \text{N}\), which matches the correct answer option.
Approach Solution -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 \( 4 \, N \).
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