Toppling and Sliding

This lesson covers the conditions under which a rigid body on an inclined surface will slide or topple. Understanding which occurs first is a classic AQA Further Mechanics topic.

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1 Overview

When a block (or other rigid body) sits on a plane inclined at angle $\theta$θ, two failure modes can occur as $\theta$θ increases:

1. Sliding: The component of weight along the slope exceeds the maximum friction force.
2. Toppling: The line of action of the weight falls outside the base of support, creating an overturning moment.

The block will do whichever occurs first (at the smaller angle).

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2 Condition for Sliding

The block slides when:

$$mg\sin\theta > \mu mg\cos\theta$$mgsinθ>μmgcosθ $$\boxed{\tan\theta > \mu}$$tanθ>μ​

The critical angle for sliding: $\theta_s = \arctan(\mu)$θs​=arctan(μ).

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3 Condition for Toppling

Consider a rectangular block of width $2a$2a and height $2h$2h, standing on the inclined plane. The centre of mass is at height $h$h above the base and $a$a from the lower edge.

The block topples about its lower edge when the vertical line through the centre of mass passes beyond this edge. The critical condition is:

$$\boxed{\tan\theta > \frac{a}{h}}$$tanθ>ha​​