Question 7: Work, Energy and Power & Circular Motion

Question

A mass of 1 kg is kept on a inclined plane with $30^{\circ}$ inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of $4 \mathrm{~m} / \mathrm{s}$. The work done by the frictional force in time 2 s is $\_\_\_\_$ J. (Take $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Accepted Answer

When the block is at rest on the incline, the forces acting on it are :
$\mathrm{F}_{\mathrm{g}}=\mathrm{mg}$ is weight acting vertically downward.
$\mathrm{N}=$ Normal force acting perpendicular to the incline.
$\mathrm{f}_{\mathrm{s}}=$ Static friction acting up the incline, preventing the block from sliding down.
Since the block is at rest relative to the incline, the frictional force must balance the component of gravity acting down the slope :
$$ \mathrm{f}_{\mathrm{s}}=\mathrm{mg} \sin 	heta $$
The mass of block is $\mathrm{m}=1 \mathrm{~kg}$, acceleration due to gravity is $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ and $	heta=30^{\circ}$
So, value of static friction is
$$ \mathrm{f}_{\mathrm{s}}=1 	imes 10 	imes \sin 30^{\circ}=10 	imes 0.5=5 \mathrm{~N} $$
The friction force is 5 N directed up along the incline.
As the entire assembly moves vertically upward with a constant velocity $\mathrm{v}=4 \mathrm{~m} / \mathrm{s}$, so in time $\mathrm{t}=2 \mathrm{~s}$, the vertical displacement $S$ is:
$$ S=v 	imes t=4 	imes 2=8 \mathrm{~m} $$
Work done is defined as $W=\vec{F} \cdot \vec{d}=F d \cos \phi$, where $\phi$ is the angle between the force and the displacement.
The angle between the friction force and the vertical is $90^{\circ}-30^{\circ}=60^{\circ}=\phi$.
So, work done by frictional force is,
$$ \mathrm{W}=\mathrm{f}_{\mathrm{s}} 	imes \mathrm{S} 	imes \cos 60^{\circ} $$
$$\Rightarrow $$ $\mathrm{W}=5 	imes 8 	imes 0.5 \mathrm{~J}$
$$\Rightarrow $$ $$ \mathrm{W}=5 	imes 4=20 \mathrm{~J} $$
Therefore, the work done by the frictional force is 20 J .
Hence, the correct option is (A).

Answer

20

Answer

25

Answer

30

Answer

10

Work, Energy and Power & Circular Motion

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