A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall. The foot of the ladder is 2 m from the wall and the top of the ladder is 7 m above the ground. The ladder is in equilibrium and is on the point of slipping. Find the coefficient of friction between the ladder and the ground. Two light inelastic strings are tied to a particle of weight 123 N. The lengths of the strings are 40 cm and 9 cm, respectively. The other ends of the strings are tied to two points 41 cm apart on a horizontal ceiling.

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A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall - Leaving Cert Applied Maths - Question 7 - 2017

Question 7

A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall. The foot of the ladder is 2 m from the wall and the to... show full transcript

Worked Solution & Example Answer:A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall - Leaving Cert Applied Maths - Question 7 - 2017

Step 1

Find the coefficient of friction between the ladder and the ground.

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Answer

To find the coefficient of friction ( $\mu$ ) between the ladder and the ground, we start by analyzing the forces acting on the ladder.

Step 1: Analyze Forces

The forces acting on the ladder include:

Weight of the ladder (W) = 210 N acting downward at the center of gravity (midpoint).
Normal reaction (R) from the ground acting upward.
Frictional force (S) acting horizontally at the base in the opposite direction to potential slipping.

Step 2: Set up Equilibrium Equations

Since the ladder is in equilibrium:

The sum of vertical forces must equal zero: $R - W = 0 \quad \Rightarrow \quad R = W = 210 N$
The moment about the base (point where ladder meets ground) must also be zero. We can use the distances to write the moment equation: $S \times 7 = W \times 1 \Rightarrow S \times 7 = 210 \times 1$ $\Rightarrow S = \frac{210}{7} = 30 N$

Step 3: Identify the Coefficient of Friction

Using the relation for friction: $\mu S = R \Rightarrow \mu = \frac{R}{S} = \frac{30}{210} = \frac{1}{7}$

Thus, the coefficient of friction is $\mu = \frac{1}{7}$ .

Step 2

Show on a diagram the forces acting on the particle.

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Answer

The diagram should include the following:

A downward force representing the weight of the particle, marked as 123 N.
Two tension forces ( $T$ and $S$ ) acting at angles to the horizontal, connecting to the points on the ceiling.
Indicate the angles of the strings with the horizontal as α and β respectively.

Step 3

Write down the two equations that arise from resolving these forces horizontally and vertically.

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Answer

For the particle in equilibrium, the forces can be resolved as follows:

Horizontal Forces Equation:

$T \cos(\alpha) = S \cos(\beta)$

Vertical Forces Equation:

$$ T \sin(\alpha) + S \sin(\beta) = 123 $.

Step 4

Solve these equations to find the tension in each of the strings.

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Answer

We use the equations established earlier:

Step 1: Substitute for $S$

From the horizontal forces equation, we can express $S$ in terms of $T$ : $S = T \frac{\cos(\alpha)}{\cos(\beta)}$ Substituting this into the vertical forces equation gives: $T \sin(\alpha) + \left( T \frac{\cos(\alpha)}{\cos(\beta)} \right) \sin(\beta) = 123$

Step 2: Solve for T

Now simplify and isolate $T$ : After some rearrangement, we will find $T = 27 N$ .

Step 3: Find $S$

Substituting the value of $T$ back into either equation will yield: $S = 120 N$

A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall - Leaving Cert Applied Maths - Question 7 - 2017

Worked Solution & Example Answer:A uniform ladder, of weight 210 N, rests on rough horizontal ground and leans against a smooth vertical wall - Leaving Cert Applied Maths - Question 7 - 2017

Find the coefficient of friction between the ladder and the ground.

Step 1: Analyze Forces

Step 2: Set up Equilibrium Equations

Step 3: Identify the Coefficient of Friction

Show on a diagram the forces acting on the particle.

Write down the two equations that arise from resolving these forces horizontally and vertically.

Horizontal Forces Equation:

Vertical Forces Equation:

Solve these equations to find the tension in each of the strings.

Step 1: Substitute for SSS

Step 2: Solve for T

Step 3: Find SSS

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Step 1: Substitute for $S$

Step 3: Find $S$