Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force $F$ at $a$ point $b$ above its centre of mass (see figure). If the coefficient of friction is $μ = 0.4$ , the maximum possible value of $100 \times \frac{b}{a}$ for box not to topple before moving is

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alok kumar singh | Contributor-Level 10

2 months ago

For no toppling

$F (\frac{a}{2} + b) \leq m g \frac{a}{2}$

$μ \frac{a}{2} + μ b \leq \frac{a}{2}$

$0.2 a + 0.4 b \leq 0.5 a$

$0.4 b \leq 0.3 a$

$b \leq \frac{3 a}{4}$

$b \leq 0.75 a$ (in limiting case)

But is not possible as maximum value of $b$ can be equal to $0.5 a$ only.

$∴ {(100 \frac{b}{a})}_{m a x} = 50.00$

For no toppling <math> <mi>F</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>b</mi> </mrow> </mrow> </mfenced> <mo>≤</mo> <mi>m</mi> <mi>g</mi> <mfrac> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> <math> <mi>μ</mi> <mfrac> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>μ</mi> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> <math> <mn>0.2</mn> <mi>a</mi> <mo>+</mo> <mn>0.4</mn> <mi>b</mi> <mo>≤</mo> <mn>0.5</mn> <mi>a</mi> </math> <math> <mn>0.4</mn> <mi>b</mi> <mo>≤</mo> <mn>0.3</mn> <mi>a</mi> </math> <math> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mfrac> <mrow> <mrow> <mn>3</mn> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math> <math> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mn>0.75</mn> <mi mathvariant="normal">a</mi> <mrow> <mi mathvariant="normal"> </mi> </mrow> </math> (in limiting case)But is not possible as maximum value of <math> <mi>b</mi> </math> can be equal to <math> <mn>0.5</mn> <mi>a</mi> </math> only. <math> <mo>∴</mo> <msub> <mrow> <mrow> <mfenced separators="|"> <mrow> <mrow> <mn>100</mn> <mfrac> <mrow> <mrow> <mi mathvariant="normal">b</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>50.00</mn> </math>

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Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is μ = 0.4 , the maximum possible value of 100 × b a for box not to topple before moving is