Given :-

• A body of MASS m slides down a smooth inclined plane having an inclination of 45° with the horizontal. It takes the body 2 seconds to reach the bottom.

To Find :-

• Time taken by the body if the coefficient of friction of the plane is 0.5

Solution :-

• Let's find the distance travelled by the body using the first case. SINCE the plane is smooth in this case, So there will be no frictional force opposing the motion of the body down the inclined plane.

• So, Acceleration with the INCLINE is g sin θ,
• Using second equation of motion,

⇒ s = UT + 1/2 at²

⇒ s = 1/2 × g sin θ × 2²

⇒ s = 2 × 10 × sin 45°

⇒ s = 20 × 1/√2

⇒ s = 10√2 units.

• In the second case, we have friction, So it will oppose the motion of body,

Here, The acceleration would become

⇒ g sin θ - μ g cos θ

⇒ 10×sin 45° - 0.5 × 10 × cos 45°

⇒ 5√2 - 5√2 / 2

⇒ ( 10√2 - 5√2 ) / 2

⇒ 5√2/2 unit/s²

• Now, Let us find the time taken by the body to cover 10√2 unit.

Using second equation of motion,

⇒ s = ut + 1/2 at²

⇒ 10√2 = 1/2 × 5√2 / 2 × t²

⇒ 10√2 = 5√2 / 4 × t²

⇒ 40√2 / 5√2 = t²

⇒ 8 = t²

⇒ t = 2√2 seconds

Hence, The body will take 2√2 seconds.