Question

A body of mass 2 kg moves in a horizontal circular path of radius 5m. In an instant its speed is \(2\sqrt5\) m/s and is increasing at the rate of 3 m/s².The magnitude of the force acting on the body at the instant is,

• A. 6N
• B. 8N
• C. 14N
• D. 10N

Answer 1

The Correct Option is D

To elaborate on the force acting on a body moving in a circular path with increasing speed: 

When a body moves in a circle and its speed is increasing, two types of accelerations act on it:

• Centripetal acceleration (\(a_c\)) – directed toward the center of the circle, responsible for changing the direction of the velocity.
• Tangential acceleration (\(a_t\)) – directed along the tangent to the circle, responsible for changing the magnitude (speed) of the velocity.

Let the mass of the body be \(m\). Then: 

• Centripetal force: \(F_c = m a_c = m \frac{v^2}{r}\)
• Tangential force: \(F_t = m a_t\)

These two forces act at right angles to each other, so the net force is given by: \[ F = \sqrt{F_c^2 + F_t^2} \] 
In the given case, suppose:

• \(F_c = 6\, \text{N}\)
• \(F_t = 8\, \text{N}\)

Then the net force is: \[ F = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = \boxed{10\, \text{N}} \] 
Correct answer: (D) 10 N

Answer 2

To determine the magnitude of the total force acting on a body moving in a horizontal circular path with increasing speed, we need to combine both:

• Centripetal force (\(F_c\)) — acts toward the center of the circle 
• Tangential force (\(F_t\)) — acts along the tangent to the circle

Step 1: Given values
\[ m = 2\, \text{kg}, \quad v = 2\sqrt{5}\, \text{m/s}, \quad r = 5\, \text{m}, \quad a_t = 3\, \text{m/s}^2 \] Step 2: Centripetal force \[ F_c = \frac{mv^2}{r} = \frac{2 \times (2\sqrt{5})^2}{5} = \frac{2 \times 20}{5} = \frac{40}{5} = 8\, \text{N} \] Step 3: Tangential force \[ F_t = ma_t = 2 \times 3 = 6\, \text{N} \] Step 4: Total force
Since \(F_c\) and \(F_t\) are perpendicular, the total force is: \[ F = \sqrt{F_c^2 + F_t^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = \boxed{10\, \text{N}} \] Conclusion:
The total force acting on the body is 10 N. 
Correct option: (D) 10 N