Question

If a ladder weighing 250N is placed against a smooth vertical wall having a coefficient of friction 0.3, then what is the maximum force of friction available at the point of contact between the ladder and the floor?
$
  (a){\text{ 75N}} \\
  (b){\text{ 50N}} \\
  (c){\text{ 35N}} \\
  (d){\text{ 25N}} \\
 $

Answer 1

- Hint: In this question consider the fact that due to the weight of the ladder there will be a reaction force working between the ladder and the floor in upward direction to balance the weight of the ladder as shown in the figure so use the concept that the maximum frictional force available at the point of contact between the ladder and the floor is the product of coefficient of friction and the reaction force working between the ladder and the floor that is ${F_f} = \mu \times {R_2}$. This will help approaching the problem.

Complete step-by-step solution -

The pictorial representation of the ladder placed against a smooth vertical wall is shown above.
The weight of the ladder is 250N and the coefficient of friction is 0.3.
Now the weight of the ladder is acting vertically downward as shown in the figure.
$ \Rightarrow W = mg = 250N$, where m = mass of the ladder and g = acceleration due to gravity.
Now due to the weight of the ladder there is a reaction force working between the ladder and the floor in upward direction to balance the weight of the ladder as shown in the figure.
$ \Rightarrow {R_2} = W = mg = 250N$
Now the coefficient of friction between the floor and the ladder is 0.3, often denoted by $\mu $
$ \Rightarrow \mu = 0.3$
So the maximum frictional force available at the point of contact between the ladder and the floor is the product of coefficient of friction and the reaction force working between the ladder and the floor.
Let the maximum frictional force be ${F_f}$ as shown in the figure.
$ \Rightarrow {F_f} = \mu \times {R_2}$
Now substitute the values we have,
$ \Rightarrow {F_f} = 0.3 \times 250 = 75N$
So the maximum frictional force working between the ladder and the floor is 75N.
This frictional force balances the reaction force $\left( {{R_1}} \right)$ working on the ladder and the smooth vertical wall so that the ladder does not slip.
So this is the required answer.
Hence option (A) is the correct answer.

Note – It is important to understand the basics of friction as it is very helpful in solving problems of this kind. Friction has mainly three types that is static friction, limiting friction and the kinetic friction. A diagram will help understand the three frictions clearly.

In the graphical representation region OE is the region where the object is stationary holding its current position and the static friction is acting upon it, the region EF is where external force has been applied and the friction is opposing the change of state, after F the object is in motion and now kinetic friction is acting upon the object.