Question

Mass m₁ moves on a slope making an angle θ with the horizontal and is attached to mass m₂ by a string passing over a frictionless pulley as shown in figure. The coefficient of friction between m₁ and the sloping surface is µ.

1. If m₂ > m₁ sin θ, the body will move up the plane.
2. If m₂ > m₁ (sin θ + µ cos θ), the body will move up the plane.
3. If m₂ < m₁ (sin θ + µ cos θ), the body will move up the plane.
4. If m₂ < m₁ (sin θ − µ cos θ), the body will move down the plane.

Answer 1

b and d

Explanation:

Let m₁, move up the plane. Different forces involved are shown in the diagram.

N = Normal reaction

f = Frictional force

T = Tension in the string

f = μN = μm₁g cos θ

For the system (m₁ + m₂) to move up

m₂g – (m₁g sin θ + f) > 0

⇒ m₂g – (m₁g sin θ + μ m₁g cos θ) > 0

⇒ m₂ > m₁ (sin θ + μ cos θ)

Hence, option b. os correct.

Let the body moves down the plane, in this case, f acts up the plane.

Hence, m₁g sin θ – f > m₂g

⇒ m₁g sin θ – μ m₁g cos θ > m₂g 

⇒ m₁ (sin θ – μ cos θ) > m₂

⇒ m₂ < m₁ (sin θ – μ  cos θ) 

Hence, option d. is correct.