An inclined plane is bent in such a way that the vertical cross-section is given by y = x^2/4 where,

Q: Why do we equate $\frac{dy}{dx}$ to $\mu$?

$\frac{dy}{dx}$ represents the slope of the tangent to the curve, which is $\tan\theta$. For a block on an inclined plane, the limiting condition for slipping is $\tan\theta = \mu$.

Q: What happens if the block is placed higher than 25 cm?

At a height greater than 25 cm, the slope of the curve ($\tan\theta$) becomes greater than $\mu$. The component of gravity pulling it down will exceed the maximum static friction, causing the block to slip.

Q: Does the mass of the block affect the maximum height?

No, the mass $m$ cancels out in the limiting equilibrium equation $mg \sin\theta = \mu mg \cos\theta$. The maximum height depends only on the shape of the curve and the coefficient of friction.

Q: Why did we convert the final answer to centimeters?

The coordinates $x$ and $y$ in the given equation are in meters (as implied by standard SI units unless specified otherwise, and the question explicitly asks for the answer in cm). Always read the required units carefully!

Q: Can we use this method for any curved surface?

Yes! As long as you know the equation of the surface cross-section, you can find the maximum height by equating the derivative $\frac{dy}{dx}$ to the coefficient of static friction $\mu$.

Question

An inclined plane is bent in such a way that the vertical cross-section is given by $y = \frac{x^2}{4}$ where, $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\mu = 0.5$, the maximum height in cm at which a stationary block will not slip downward is .......... cm.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a giant, smooth parabolic bowl. The shape of this bowl is given by the mathematical equation $$y = \frac{x^2}{4}$$. Now, you place a block on this curved surface. Will it stay there, or will it slide down?

That depends entirely on two things: how steep the surface is at that specific point, and how much friction is available to hold it back. The problem states that the coefficient of static friction, $$\mu$$, is $$0.5$$. Our goal is to find the maximum height $$h$$ (or $$y$$) where the block can sit peacefully without slipping.

### The Master Equation

For any object resting on an inclined plane, the condition for it to just begin slipping (the verge of motion) is when the angle of inclination $$	heta$$ reaches the **angle of repose**. Mathematically, this is expressed as:

$$	an	heta = \mu$$

But our surface isn't a straight ramp; it's a curve! How do we find the angle of inclination on a curve? This is where calculus comes to the rescue. The slope of the tangent to any curve at a given point is exactly equal to its derivative, $$\frac{dy}{dx}$$. Therefore, we can write:

$$\frac{dy}{dx} = 	an	heta = \mu$$

### Executing the Calculus

Let's take our equation $$y = \frac{x^2}{4}$$ and differentiate it with respect to $$x$$:

$$\frac{dy}{dx} = \frac{d}{dx}\left(\frac{x^2}{4}ight) = \frac{2x}{4} = \frac{x}{2}$$

Now, we equate this slope to our given coefficient of friction, $$\mu = 0.5$$:

$$\frac{x}{2} = 0.5$$

Solving for $$x$$, we get:

$$x = 1 	ext{ m}$$

### Final Calculation

We have found the horizontal position $$x$$ where the block is on the verge of slipping. But the question asks for the **maximum height**, which corresponds to the $$y$$-coordinate. Let's plug our $$x$$ value back into the original equation of the parabola:

$$y = \frac{(1)^2}{4} = 0.25 	ext{ m}$$

**Watch out for the trap!** The question specifically asks for the answer in **centimeters**. A common silly mistake is to write $$0.25$$ as the final answer. We must convert meters to centimeters:

$$h = 0.25 	ext{ m} 	imes 100 	ext{ cm/m} = 25 	ext{ cm}$$

And there we have it! The maximum height at which the block will not slip is **$$25 	ext{ cm}$$**.

Enter Numerical Value:

Visualized Solution

Visual Anchor

Logic Bridge

Raw Setup

Atomic Compute

Atomic Compute

Atomic Compute

Final Answer

The Way Forward

The Sigma Insight: Static and Kinetic Friction

Analyzing the Setup

The Master Equation

Executing the Calculus

Final Calculation

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