Question

The 6.50 m long uniform ladder has a mass of 13.0 kg (weight: 130 N). Let µs = 0.25 and µk = 0.20

A)         BEFOREthe person climbs onto the ladder, it is placed against the wall and makes an angle of 59.49° with the floor.  Does the ladder slip?  SHOW WHY/WHY NOT

B)         Then the ladder is moved until it makes an angle of 53.13° with the floor.  How far up the ladder can a 40.0 kg (400 N) person climb?

C)         For what range of angles can the 40.0 kg person climb to the top of the ladder?

Answer 1

(a) The forces actiong on the laddder are shown in the figure below.

For the system to be in equilibrium the forces and the torques acting on the opposite directions should be equal. Here = 59.49.

At equilibrium, forces on x-direction,

Taking forces on the y-direction,

Taking torque assuming A as the point of rotation,

Given the coefficient of statis friction, = 0.25. The maximum value of frictional force that the floor can exert is

But in this case,

So the ladder will slip off.

(b) Now = 53.13. Assume that the weight w =400N person climebed be x. The forces acting on the system are given in the figure beow.

At equilibrium, forces on x-direction,

Taking forces on the y-direction,

So,

Taking torque assuming A as the point of rotation,

The person can climb 1.81m up the ladder.

(c) If the person climb fully up the ladder, hen x = l. Let that angle be . Now taking torque assuming A as the point of rotation,

Dividing throughout by cos

When = 74.10 he can fully climb up the ladder. This is the minimum angle which he can climb up the ladder.

When the ladder starts to slip (given = 0.20)

Therefore,

At this angle the ladder will slip. So the maximum angle he can climb fully up the ladder should be less than 77.17 degree. Therefore the range of angles the person can climb up to the top of the ladder is