A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C: The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α, where sin α = \frac{4}{5}, as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. Using the model, a) show that the magnitude of the force exerted on the ladder by the rail at C is \frac{9Mg}{25}. b) Hence, or otherwise, find the value of μ.

A-Level Edexcel Maths Mechanics Forces: A ladder AB has mass M and lengt

A ladder AB has mass M and length 6a - Edexcel - A-Level Maths Mechanics - Question 4 - 2020 - Paper 1

Question 4

A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C: ... show full transcript

Worked Solution & Example Answer:A ladder AB has mass M and length 6a - Edexcel - A-Level Maths Mechanics - Question 4 - 2020 - Paper 1

Step 1

a) Show that the magnitude of the force exerted on the ladder by the rail at C is \frac{9Mg}{25}

96%

114 rated

Answer

To find the force exerted at point C, we start by taking moments about point A. The equation we can use is: $N \cdot \frac{4a}{\sin \alpha} = Mg \cdot 3a \cdot \cos \alpha$ where N is the force exerted by the rail at C, and we have used the length parameters from the problem.

Substituting sin α = \frac{4}{5} into the equation and rearranging, we find: $N = \frac{9Mg}{25}$ This confirms the required formula for the force at C.

Step 2

b) Hence, or otherwise, find the value of μ.

99%

104 rated

Answer

Next, we resolve the forces acting on the ladder both horizontally and vertically. The equations we set up are:

For horizontal forces: $F - \frac{9Mg}{25} \cdot \sin \alpha = 0$ For vertical forces: $R + F \cdot \cos \alpha - Mg = 0$ where F is the frictional force and R is the normal reaction force at A.

Using the earlier derived equation for N: $N = \frac{9Mg}{25}$ So, we can substitute N in terms of friction as: $F = \mu R$

From these equations, we perform elimination of R and F to express μ: $\mu = \frac{F}{R} = \frac{36Mg}{125} \cdot \frac{5}{Mg}$

Carrying out the simplifications leads us to: $\mu = \frac{18}{49}$ Thus, the coefficient of friction between the ladder and ground is \mu \approx 0.367.

A ladder AB has mass M and length 6a - Edexcel - A-Level Maths Mechanics - Question 4 - 2020 - Paper 1

Worked Solution & Example Answer:A ladder AB has mass M and length 6a - Edexcel - A-Level Maths Mechanics - Question 4 - 2020 - Paper 1

a) Show that the magnitude of the force exerted on the ladder by the rail at C is \frac{9Mg}{25}

b) Hence, or otherwise, find the value of μ.

Join the A-Level students using SimpleStudy...