A block of mass and another mass , are placed together (see figure) on an inclined plane with angle of inclination . Various values of are given in List-I. The coefficient of friction between the block and plane is always zero. [2014]

Question

A block of mass $m_{1} = 1 kg$ and another mass $m_{2} = 2 kg$ , are placed together (see figure) on an inclined plane with angle of inclination $θ$ . Various values of $θ$ are given in List-I. The coefficient of friction between the block $m_{1}$ and plane is always zero. The coefficient of static and dynamic friction between the block $m_{2}$ and the plane are equal to $μ = 0.3$ . In List-II expressions for the friction on block $m_{2}$ are given. Match the correct expression of the friction in List-II with the angles given in List-I, and choose the correct option. The acceleration due to gravity is denoted by $g$ .

Useful information: $\tan (5.5 °) \approx 0.1; \tan (11.5 °) \approx 0.2; \tan (16.5 °) \approx 0.3$ [2014]

	List - I		List - II
P.	$θ = 5 °$	1.	$m_{2} g \sin θ$
Q.	$θ = 10 °$	2.	$(m_{1} + m_{2}) g \sin θ$
R.	$θ = 15 °$	3.	$μ m_{2} g \cos θ$
S.	$θ = 20 °$	4.	$μ (m_{1} + m_{2}) g \cos θ$

Code:

Useful information: $\tan (5.5 °) \approx 0.1; \tan (11.5 °) \approx 0.2; \tan (16.5 °) \approx 0.3$ [2014]

	List - I		List - II
P.	$θ = 5 °$	1.	$m_{2} g \sin θ$
Q.	$θ = 10 °$	2.	$(m_{1} + m_{2}) g \sin θ$
R.	$θ = 15 °$	3.	$μ m_{2} g \cos θ$
S.	$θ = 20 °$	4.	$μ (m_{1} + m_{2}) g \cos θ$

Code:

Smart Achievers · Accepted Answer

(4)

Block will not slip or will be at rest if $(m_{1} + m_{2}) g \sin θ \leq μ m_{2} g \cos θ$

$\tan θ \leq \frac{μ m_{2} g}{(m_{1} + m_{2}) g}$

$\Rightarrow \tan θ \leq \frac{μ m_{2}}{m_{1} + m_{2}}$

$\Rightarrow \tan θ \leq \frac{0.3 \times 2}{1 + 2} \leq \frac{1}{5}$

$\Rightarrow \tan θ \leq 0.2 i.e., θ \leq 11.5 °$

i.e., If the angle $θ < 11.5 °$ , the frictional force is less than $μ N_{2} = μ m_{2} g = 0.3 \times 2 \times g = 0.6 g$

and is equal to $(m_{1} + m_{2}) g \sin θ$ .

Blocks will not slip on the inclined plane and friction is static.

At $θ > 11.5 °$ , the bodies start moving on the inclined plane and friction is kinetic and equal to $μ m_{2} g \cos θ$ .

Solution

1 P-1, Q-1, R-1, S-3

2 P-2, Q-2, R-2, S-3

3 P-2, Q-2, R-2, S-4

4 P-2, Q-2, R-3, S-3

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