A ladder 10-m long and weighing 50 N rests against a smooth vertical wall. If the ladder is just on the verge of slipping when it makes a 50 degree angle with the ground, find the coefficient of static friction between the ladder and the ground.

I want the work to be shown

Well, well, looks like we have a slippery situation on our hands! Let's see if we can solve this with a touch of humor.

So, we have a ladder that's on the verge of slipping. It's probably thinking, "I'm so inclined to take a leap!"

First, let's break down the forces at play here. We have the weight of the ladder, which is 50 N, acting downwards. Now, imagine the ladder as a comedian trying to keep its balance on a slippery stage.

When the ladder is on the verge of slipping, we know that the friction force must be equal to the maximum value of static friction, which is given by the coefficient of static friction (μ) multiplied by the normal force.

In this case, the normal force is acting perpendicular to the inclined ladder, and the friction force is acting parallel to the inclined ladder. They balance each other out to prevent any slipping shenanigans.

Now, the normal force can be resolved into two components: one perpendicular to the wall and one parallel to the wall. All the vertical component does is apply some pressure, trying to keep the ladder in place. But it's the parallel component that keeps the comedy ladder from slipping.

The parallel component of the normal force can be found by multiplying the weight of the ladder (50 N) by the sine of the angle (50 degrees) with the ground. The sine of 50 degrees might be like a punchline that's hard to remember, but it's approximately 0.766.

So, now we have the parallel component of the normal force, which is opposing the tendency to slip. And coincidentally, it's equal to the maximum static friction force, which is also acting parallel to the ladder.

Therefore, the maximum static friction force on the ladder is 50 N multiplied by 0.766, which is approximately 38.3 N.

Finally, we can plug this value into the equation for static friction force:

Static friction force = coefficient of static friction (μ) × normal force

We already know the maximum static friction force, which is 38.3 N. Dividing this by the normal force, which is the weight of the ladder (50 N) multiplied by the cosine of the angle (50 degrees), we get the coefficient of static friction.

So, in our hilarious equation:

38.3 N = μ × (50 N × cos(50 degrees))

Calculating this out, we find that the coefficient of static friction (μ) is approximately 0.766.

Thus, the slippery clown ladder has a coefficient of static friction of around 0.766. It seems like it's managing to clown around without slipping too much!

Remember, when in doubt, add a touch of comedy to calculations. It never hurts to make them a little less dull!

To find the coefficient of static friction between the ladder and the ground, we need to consider the forces acting on the ladder.

Let's analyze the forces acting on the ladder:

1. Weight of the ladder: The weight of the ladder acts vertically downward and is given as 50 N.

2. Normal force: The normal force acts perpendicular to the ground and opposes the weight of the ladder. It is equal in magnitude to the weight of the ladder, which is 50 N.

3. Frictional force: The frictional force acts parallel to the ground and opposes the tendency of the ladder to slip. We need to find this force.

Since the ladder is just on the verge of slipping, the frictional force is at its maximum value. This maximum value is calculated using the equation:

Frictional force (Ff) = coefficient of static friction (μ) * normal force (N)

We know the normal force is 50 N, and we need to find the coefficient of static friction. We can solve for μ:

Ff = μN

50 N = μ * 50 N

Dividing both sides of the equation by 50 N:

μ = 50 N / 50 N

Therefore, the coefficient of static friction between the ladder and the ground is equal to 1.

To find the coefficient of static friction between the ladder and the ground, we can analyze the forces acting on the ladder.

Let's consider the forces acting on the ladder:
1. Gravitational force (weight of the ladder): This force acts vertically downward and can be calculated by multiplying the mass of the ladder by the acceleration due to gravity (N/kg).
2. Normal force: This force acts perpendicular to the wall and opposes the gravitational force. Since the ladder is on the verge of slipping, the normal force will be equal in magnitude and opposite in direction to the perpendicular component of the gravitational force.
3. Frictional force: This force acts parallel to the ground and opposes the motion or tendency of motion between the ladder and the ground.

In this problem, we need to consider the forces acting on the ladder in equilibrium, because it is just on the verge of slipping.

We know that the ladder makes a 50-degree angle with the ground. Therefore, the angle between the ladder and the wall is (90 degrees - 50 degrees) = 40 degrees.

Let's break down the forces acting on the ladder:
1. Gravitational force (weight): This force acts vertically downward and has a magnitude given by:
Weight = mass x acceleration due to gravity = m x g

2. Perpendicular component of the weight: This component acts perpendicular to the wall. We can find the magnitude of this component using trigonometry:
Perpendicular component of weight = Weight x cos(angle between ladder and ground)

3. Frictional force: This force acts parallel to the ground. Since the ladder is at the verge of slipping, the frictional force (Ff) can be calculated as the product of the coefficient of static friction (μs) and the perpendicular component of the weight:
Ff = μs x perpendicular component of weight

4. Normal force: This force acts perpendicular to the wall and balances the perpendicular component of the weight:
Normal force = perpendicular component of weight

Now, let's plug in the given values:
Length of the ladder (L) = 10 m
Weight of the ladder (Weight) = 50 N
Angle with the ground (θ) = 50 degrees

Step 1: Calculate the perpendicular component of the weight:
Perpendicular component of weight = Weight x cos(θ)

Step 2: Calculate the frictional force:
Ff = μs x perpendicular component of weight

Step 3: Calculate the normal force:
Normal force = perpendicular component of weight

Since the ladder is in equilibrium, the frictional force must be equal in magnitude to the normal force. Hence, we can write:
Ff = Normal force

Therefore, the coefficient of static friction (μs) can be found by dividing the frictional force by the perpendicular component of the weight:
μs = Ff / perpendicular component of weight

Now, let's calculate the values to find μs:

Step 1: Calculate the perpendicular component of the weight:
Perpendicular component of weight = 50 N x cos(50 degrees)

Step 2: Calculate the frictional force:
Ff = μs x perpendicular component of weight (which we found in Step 1)

Step 3: Calculate the coefficient of static friction:
μs = Ff / perpendicular component of weight (which we found in Step 1)

By evaluating these expressions, we can determine the coefficient of static friction between the ladder and the ground.

The upper wall exerts only a normal force on the ladder -- no friction force. The normal force there (F1) is such that

F1*L sin 50 = W*L/2* cos 50
F1 = (1/2)*W tan 50 = 0.596 W
The normal force at the base equals the weight W, since the other contact point is smooth and frictionless. If it is just about to slip, the maximum static friction force at the base (Ff,max) equals the normal force at the top, which I called F1.

Ff,max = mus*W = F1 = W/2 tan 50

mus = F1/W = (1/2) tan 50 = 0.596
is the static friction coefficient