(i) The acceleration of m in the S’ frame is (A) (M + m) gsinθ/ m + msin^2 θ (B) (M + m)g sinθ/ m + Msinθ

Question

Comprehension Type :

Paragraph: A block of mass m slides down a smooth incline of mass M and length l, solely as a result of the force of gravity. The incline is placed on a smooth horizontal table as shown in Figure. Let us denote the coordinate system relative to the table as S₁ and the coordinate system relative to the incline as S_\(\phi\) 

(i) The acceleration of m in the S’ frame is

(A) \(\frac{(M + m) g sin\theta}{M + m sin^2 \theta}\)

(B) \(\frac{(M + m) g sin\theta}{m + M sin^2 \theta}\)

(C) \(\frac{(M - m) g sin\theta}{M + m sin^2 \theta}\)

(D) \(\frac{(M + m) g sin\theta}{M + m sin \theta}\) 

(ii) The acceleration of the incline in the S frame

(A) \(\left(\frac{mg\, sin\theta\, cos\theta}{M + m sin^2 \theta} \right)\) 

(B) -\(\left(\frac{mg\, sin\theta\, cos\theta}{M + m sin^2 \theta} \right)\) 

(C) \(\left(\frac{mg\, sin\theta\, cos\theta}{M + m sin^2 \theta} \right)\) 

(D) -\(\left(\frac{mg\, sin\theta\, cos\theta}{M + m sin^2 \theta} \right)\) 

(iii) The force exerted by the small m on the wedge of mass M

(A) mg cos\(\theta\)

(B) \(\frac{Mmg}{M + m sin^2 \theta}\) 

(C) \(\frac{mg}{cos \theta}\) 

(D) None

(iv) At what acceleration a_x (in the S frame) must the incline be accelerated to prevent m from sliding

(A) - g tan \(\theta\)

(B) + g tan\(\theta\)

(C) \(-\frac{g\, tan \theta}{2}\)

(D) \(\frac{g\, tan \theta}{2}\)

Answer 1

(i) (A) \(\frac{(M + m)g\sin \theta}{M+m\sin^2\theta}\)

(ii) (B) \(\left(\frac{mg \sin\theta \cos\theta}{M+ m \sin^2\theta}\right)\)

(iii) (B) \(\frac {Mmg}{M+m\sin^2\theta}\)

(iv) (B) \(+ g\tan \theta\)

In s’ frame;

This is the acceleration of the block ‘m’ with respect to the incline.

Force exerted by the mass ‘m’ on wedge is ‘N’.

We can find this by; equation (iii)

∴ N + ma sin θ = mg cos θ

⇒ N = mg cos θ – ma sin θ

Now in this question; the downward component of mg sin θ has to be balanced.

∴ max cos θ = mg sin θ

a_x = g tan θ in positive x direction.

Answer 2

(i) (A) \(\frac{(M + m) g sin\theta}{M + m sin^2 \theta}\)

(ii) (B) \(\left(\frac{mg\, sin\theta\, cos\theta}{M + m sin^2 \theta} \right)\)

(iii) (B) \(\frac{Mmg}{M + m sin^2 \theta}\)

(iv) (B)   + g tan\(\theta\)