7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground. The coefficient of friction at the ground is $rac{1}{4}$ and at the wall is $rac{1}{2}$. The ladder makes an angle $\alpha$ with the horizontal and is in a vertical plane which is perpendicular to the wall. Find $\tan \alpha$. (b) Two equal uniform rods AB and BC smoothly jointed at B are in equilibrium with the end C resting on a rough horizontal surface. The end A is held above the surface. The rod AB is horizontal and the rod BC is inclined at an angle of $45^{\circ}$ to the horizontal. If C is on the point of slipping find the coefficient of friction.

Leaving Cert Applied Maths Statics: 7. (a) One end of a uniform lad

7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground - Leaving Cert Applied Maths - Question 7 - 2008

Question 7

7.-(a)-One-end-of-a-uniform-ladder,-of-weight-$W$,-rests-against-a-rough-vertical-wall,-and-the-other-end-rests-on-rough-horizontal-ground-Leaving Cert Applied Maths-Question 7-2008.png

7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground. The coefficient of frict... show full transcript

Worked Solution & Example Answer:7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground - Leaving Cert Applied Maths - Question 7 - 2008

Step 1

Find $\tan \alpha$

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Answer

To determine $\tan \alpha$ , we analyze the forces and moments acting on the ladder.

Equilibrium of Forces:
The sum of vertical forces must equal zero: $R + R_{t} = W$
where $R$ is the normal reaction at the ground and $R_{t}$ is the normal reaction at the wall.
Moment about Point O:
Taking moments about the base of the ladder: $R \cdot (\sin \alpha) + W \cdot (\frac{\xi}{(\cos \alpha)}) = R_{t} \cdot (\cos \alpha)$

Here, $\xi$ represents the distance from the wall to the point where the ladder touches the ground.
Equating Moments:
Rearranging gives us: $R \tan \alpha + \frac{1}{2}(R_{t} + R) = R_{t}$
which simplifies to:
$\frac{1}{2} R_{t} \tan \alpha + \frac{1}{2}R = R_{t}$
Solving for $\tan \alpha$ :
The final derivation leads to: $\tan \alpha = \frac{7}{4}$

Step 2

Find the coefficient of friction if C is on the point of slipping

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Answer

For the scenario of rods AB and BC:

Equilibrium Condition:
As rod AB is held horizontal and rod BC is inclined, we analyze moments about point A: $W + W \left(\frac{l}{\sqrt{2}} + \mu R \left(\frac{2l}{\sqrt{2}}\right)\right) = R \left(2 \cdot \frac{l}{\sqrt{2}}\right)$
Moments about B:
Write the moment equation about B: $W \left(\frac{1}{\sqrt{2}}\right) + \mu R \left(\frac{2}{\sqrt{2}}\right) = W + 2 \mu R$
Solving gives: $R \cdot 2 = W + 2R$
On simplifying leads to: $R = \frac{3W}{2}$
Final Expression for the Coefficient of Friction:
Plugging in to find $\mu$ gives us: $\mu = \frac{2}{3}$

7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground - Leaving Cert Applied Maths - Question 7 - 2008

Worked Solution & Example Answer:7. (a) One end of a uniform ladder, of weight $W$, rests against a rough vertical wall, and the other end rests on rough horizontal ground - Leaving Cert Applied Maths - Question 7 - 2008

Find $\tan \alpha$

Find the coefficient of friction if C is on the point of slipping

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