Q #2: (only other one I'm stuck on...)

A 10-kg ladder 2.5m long rests agaist a frictionless wall with its base on the floor 80cm from the wall. How much force does the top of the ladder exert on the wall?
I've found answers to similar questions on this site, but not close enough for me to figure this out. Thanks.

A = angle from floor to ladder

get it from cos A = 0.80/2.5

Take moments around base of ladder

F * distance of top of ladder above floor
= Weight of ladder * half distance of foot of ladder from wall

F * 2.5 sin A = 10*9.8 * (2.5/2) cos A

The answer is supposed to be 16.5N. I can't get it. Is angle A, 100deg? I took 0.80/2.5 = .32 and then cos (.32) = .9999deg.

And then in the final equation, for
sin A I got sin (100deg) = 98.4. This doesn't seem right.

To determine the force exerted by the top of the ladder on the wall, you can use the principles of equilibrium, specifically the torque balance. Torque is the measure of a force's tendency to rotate an object around an axis.

Let's break down the problem and go step by step:

Step 1: Identify the forces acting on the ladder.
In this case, the only external force acting on the ladder is the weight (mg) acting vertically downward from its center of mass. Here, 'm' is the mass of the ladder (10 kg), and 'g' is the acceleration due to gravity (9.8 m/s²).

Step 2: Find the distance between the point of rotation and the force.
The force we are interested in is the force exerted by the top of the ladder on the wall. This force acts at a perpendicular distance from the point of rotation, which is the bottom of the ladder. The given length of the ladder (2.5 m) and the distance from the wall to the base (80 cm) can help us find this distance.

Since the ladder is resting on the floor, the length of the ladder must be equal to the sum of the horizontal distance from the wall to the bottom of the ladder (80 cm) and the horizontal distance from the bottom of the ladder to the top of the ladder (unknown). So we have:

Length of Ladder = Distance from wall to base + Distance from base to top
2.5 m = 0.8 m + Distance from base to top

Therefore, Distance from base to top = 2.5 m - 0.8 m = 1.7 m

So the distance from the point of rotation (bottom of the ladder) to the force (top of the ladder) is 1.7 m.

Step 3: Determine the force exerted by the top of the ladder on the wall.
To maintain equilibrium, the torque exerted by the weight of the ladder (mg) must be balanced by the torque exerted by the force at the top of the ladder. Mathematically, this can be expressed as:

Torque due to weight = Torque due to force

The torque due to the weight of the ladder is given by the product of the weight (mg) and the perpendicular distance from the point of rotation (bottom of the ladder) to the line of action of the weight force.

The torque due to the force exerted by the top of the ladder on the wall is given by the product of the force and the perpendicular distance from the point of rotation (bottom of the ladder) to the line of action of the force.

In this case, since the ladder is frictionless and not moving, the torques are balanced and we have:

mg * Distance from bottom to bottom of ladder = Force * Distance from bottom to top of ladder

Substituting the values we found:
(10 kg * 9.8 m/s²) * 0.8 m = Force * 1.7 m

Simplifying the equation:
Force = ((10 kg * 9.8 m/s²) * 0.8 m) / 1.7 m

Calculating the force:
Force = 45.88 N

Therefore, the top of the ladder exerts a force of approximately 45.88 Newtons on the wall.