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A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in Fig. Match the following (most appropriate choice) :

Column I Column II
(a) mg/4 <F < mg/2 (i) Cube will move up.
(b) F > mg/2 (ii) Cube will not exhibit motion.
(c) F > mg (iii) Cube will begin to rotate and slip at A.
(d) F > mg/4 (iv) Normal reaction effectively at a/3 from A, no motion.

1 Answer

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Best answer

Consider the given diagram –

Moment of force due to F at A (anticlockwise)

\(τ_1=\vec{AB}\times \vec{F}=a\times \vec{F} \)

Moment of force due to mg at A (Clockwise),

τ2 = mg × \(\frac{a}{2}\)

(i) Cube will not move (exhibit motion), if τ1 = τ2

\(a\times \vec{F}\) = mg\(\frac{a}{2}\) or \(\vec{F}\)\(\frac{mg}{2}\) so, (a) → (ii)

(ii) Cube will rotate, if τ1 > τ2

Fa > mg\(\frac{a}{2}\) or F > \(\frac{mg}{2}\) So, (b) → (iii)

(iii) Let Normal reaction is acting at a/3 from point A,

mg× \(\frac{a}{3}\) = F × a or F = \(\frac{mg}{3}\)\(\frac{mg}{4}\) So, (c) → (i)

(iv) \(\frac{mg}{4}<\frac{mg}{3}\) if, so due to F, block will not move, so, (d) → (iv)

∴ (a) → (i), (b) → (iii), (c) → (i), (d) → (iv)

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