Question

The magnitude and direction of the acceleration produced in a body of mass 5 kg when two mutually perpendicular forces 8 N and 6 N act on it, are respectively:

• A. 2 m s⁻²; $\tan^{-1}(4/3)$ with 8 N force
• B. 2 m s⁻²; $\tan^{-1}(3/4)$ with 8 N force
• C. 2 m s⁻²; $\tan^{-1}(3/4)$ with 6 N force
• D. 20 m s⁻²; $\tan^{-1}(4/3)$ with 8 N force

Hint:

Always associate the denominator in the $\tan \theta$ formula with the force from which you are measuring the angle. Angle with 8 N $\rightarrow$ 8 is in the denominator.

Answer 1

The Correct Option is B

Step 1: Understanding the Topic:
This problem deals with "Laws of Motion" and the vector nature of force. According to Newton's Second Law, the acceleration of an object is determined by the net (resultant) force acting upon it. When multiple forces are involved, we cannot simply add their magnitudes; we must use vector addition to find the total force vector.
Step 2: Key Formulas and Approach:
For two forces $F_1$ and $F_2$ acting perpendicularly:
Resultant Force Magnitude: $F_{net} = \sqrt{F_1^2 + F_2^2}$.
Newton's Second Law: $a = F_{net} / m$.
Direction ($\theta$) relative to force $F_1$: $\tan \theta = F_2 / F_1$.

Step 3: Detailed Explanation:

Determine the Net Force: The forces 8 N and 6 N are mutually perpendicular (at $90^\circ$ to each other). We use the Pythagorean theorem for vectors: \[ F_{net} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ N} \]
Calculate the Acceleration: With a mass of 5 kg, the magnitude of acceleration is: \[ a = \frac{F_{net}}{m} = \frac{10 \text{ N}}{5 \text{ kg}} = 2 \text{ m/s}^2 \]
Determine the Direction: We need the angle $\theta$ that the resultant force (and thus acceleration) makes with the 8 N force. \[ \tan \theta = \frac{\text{Opposite Side (6 N)}}{\text{Adjacent Side (8 N)}} = \frac{6}{8} = \frac{3}{4} \] \[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]
The acceleration vector points in the same direction as the net force, making an angle of $\tan^{-1}(3/4)$ with the 8 N force vector.
Step 4: Final Answer:
The acceleration is 2 m s⁻² at an angle of $\tan^{-1}(3/4)$ with the 8 N force.