A particle is projected along a line of greatest slope up a rough plane inclined at an angle of o 45° with the horizontal. If the coefficient of friction is 1/2 , then the retardation is

The correct answer is (g/√2)(1 + 1/2).

We have a particle projected up a rough inclined plane. Let's identify all the forces and their directions carefully before applying Newton's second law.

Given Information:
- Angle of inclination: θ = 45°
- Coefficient of friction: μ = 1/2
- The particle moves up the slope (so friction acts down the slope, opposing motion)

Step 1: Find the Normal Reaction (N)

The component of gravity perpendicular to the inclined surface is balanced by the normal reaction:

N=mgcos45°=\frac{\mathrm{mg}}{\sqrt{2}}

Step 2: Find the Friction Force (f)

Since the particle is moving up the slope, kinetic friction acts down the slope:

f=\mu N=\frac{1}{2}\times \frac{\mathrm{mg}}{\sqrt{2}}=\frac{\mathrm{mg}}{2\sqrt{2}}

Step 3: Find the Gravity Component Along the Plane

The component of gravity acting down the slope (opposing upward motion):

mgsin45°=\frac{\mathrm{mg}}{\sqrt{2}}

Step 4: Calculate Total Retarding Force

Both the gravity component and friction act down the slope when the particle moves up. So the total retarding force is:

F_{\mathrm{ret}}=mgsin45°+\mu mgcos45°

F_{\mathrm{ret}}=\frac{\mathrm{mg}}{\sqrt{2}}+\frac{1}{2}\times \frac{\mathrm{mg}}{\sqrt{2}}

Step 5: Factor and Simplify

Factor out \frac{\mathrm{mg}}{\sqrt{2}} from both terms:

F_{\mathrm{ret}}=\frac{\mathrm{mg}}{\sqrt{2}}\left(1+\frac{1}{2}\right)

Step 6: Find Retardation (a)

Dividing the total retarding force by mass m:

a=\frac{F_{\mathrm{ret}}}{m}=\frac{g}{\sqrt{2}}\left(1+\frac{1}{2}\right)

This can also be written numerically as:

a=\frac{g}{\sqrt{2}}\times \frac{3}{2}=\frac{3}{g}

Key Physical Insight: The retardation is greater when the particle moves up compared to when it moves down, because during upward motion, both gravity and friction oppose the motion. During downward motion, they act in opposite directions, reducing the net force. This is why objects decelerating up a rough slope stop sooner than expected.

Therefore, the retardation of the particle is:

\frac{g}{\sqrt{2}}\left(1+\frac{1}{2}\right)