The plank is 3.6 metres long. It is being used as a lever to lift a 40-kilogram box. This represents a load of 400 newtons.

(a) If the fulcrum were placed in the centre of the plank, what downward effort would the man on the right need to apply to raise the box?
(b) If the man wants to push down on the end of the plank with an effort only of 200 newtons:
(i) how far along the plank should the fulcrum be
(ii) what would the mechanical advantage of the lever be?

To solve this problem, we can use the principles of lever mechanics and the concept of torque.

(a) If the fulcrum were placed in the center of the plank, the lever is balanced, meaning that the clockwise and counterclockwise torques are equal. The equation for torque is given by:

Torque = Force x Distance

We know that the plank is 3.6 meters long, so the distance from the fulcrum to the box is 1.8 meters on both sides. Let's assume that the man's downward effort on the right side is represented by 'F.'

On the right side of the fulcrum, the torque exerted by the man can be calculated as follows:

Torque(man) = F x 1.8

On the left side of the fulcrum, the torque exerted by the box is given by:

Torque(box) = 400 N x 1.8

Since the lever is balanced, the torques on both sides are equal. Therefore:

F x 1.8 = 400 N x 1.8

To solve for F, divide both sides of the equation by 1.8:

F = (400 N x 1.8) / 1.8

F = 400 N

Therefore, the man on the right needs to apply a downward effort of 400 newtons to raise the box if the fulcrum is placed in the center of the plank.

(b)(i) If the man wants to push down on the end of the plank with an effort of 200 newtons, we need to determine the position of the fulcrum. Let's assume the distance from the fulcrum to the end of the plank is represented by 'd.'

The torque exerted by the man on the right side (200 N) is given by:

Torque(man) = 200 N x d

The torque exerted by the box on the left side is still determined by:

Torque(box) = 400 N x 1.8

Since the lever is still balanced, the torques on both sides are equal:

200 N x d = 400 N x 1.8

To solve for 'd', divide both sides of the equation by 200 N:

d = (400 N x 1.8) / 200 N

d = 3.6 meters

So, the fulcrum should be placed 3.6 meters from the end of the plank.

(b)(ii) The mechanical advantage of a lever is calculated by dividing the distance from the fulcrum to the load (on the left side) by the distance from the fulcrum to the effort (on the right side). In this case:

Mechanical Advantage = Distance(box) / Distance(man)

Mechanical Advantage = 1.8 meters / 3.6 meters

Mechanical Advantage = 0.5

Therefore, the mechanical advantage of the lever in this scenario would be 0.5.