What Are the Equations to Solve a Ladder's Minimum Angle Problem with Friction?

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This is known as the "right-hand rule" and is used to find the direction of the torque.In summary, the problem involves finding the minimum angle (theta) at which a ladder with length L and weight W, resting against a wall, can make with the floor without slipping. The coefficient of static friction between the ladder and the floor and between the ladder and the wall is given as mu=0.5. To solve the problem, one needs to use the equations for the sum of torques, sum of forces in the y-direction, and sum of forces in the x-direction, which involve the normal forces from the floor and the wall, friction forces, and the ladder's weight. The torque point is set at the ladder
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wolfgarre
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Need help solving/setting up this problem for Physics:

A ladder of length L and weight W rests against a wall. The coefficient of static friction between the ladder and the floor and between the ladder and the wall is mu=0.5 What is the minimum angle the ladder can make with the floor (theta) without slipping?

What i know:
sum of torques = 0
sum of forces in y dir = 0
sum of forces in x dir = 0
forces in y dir = normal force from floor, friction with wall (mu times normal force from wall),and weight (W)
forces in x dir = friction with floor (mu times normal force from floor), normal force from wall

I have set the torque point at the ladder's contact with the floor and know there will be five equations to deal with.
If anyone could help me find these equations or help me set the problem up i'd appreciate it.

Jeff
 
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  • #2
The torque from a force is r sin α where r is the length of the level arm to the point where the force is applied, and α is the angle that the force makes with the arm.
 
  • #3


To solve this problem, we will use the equations for rotational equilibrium and the equations for static friction. Let's start by setting up the problem and defining our variables:

- L: length of the ladder
- W: weight of the ladder
- mu: coefficient of static friction
- theta: angle between the ladder and the floor

We can start by drawing a free body diagram of the ladder, which will help us visualize the forces acting on it. The ladder is in equilibrium, so the sum of all forces and torques acting on it must equal zero.

First, let's consider the forces acting in the y-direction (perpendicular to the floor). These include the normal force from the floor, the normal force from the wall, and the weight of the ladder. We can write an equation for the sum of forces in the y-direction:

ΣFy = Nf + Nw - W = 0

where Nf is the normal force from the floor, Nw is the normal force from the wall, and W is the weight of the ladder. We can also use the equation for static friction to relate the normal force and the coefficient of friction:

Nf = mu*Nw

Next, let's consider the forces acting in the x-direction (parallel to the floor). These include the friction force from the floor and the normal force from the wall. We can write an equation for the sum of forces in the x-direction:

ΣFx = f - Nw = 0

where f is the friction force from the floor. We can also use the equation for static friction to relate the friction force and the normal force:

f = mu*Nw

Now, let's consider the torques acting on the ladder. We will choose the point of contact between the ladder and the floor as our pivot point. The forces acting on the ladder that will create a torque are the weight of the ladder and the friction force from the wall. We can write an equation for the sum of torques:

Στ = -W*L*sin(theta) + f*L*cos(theta) = 0

where L is the length of the ladder and theta is the angle between the ladder and the floor. We can also substitute in our expression for f from the equation for the sum of forces in the x-direction:

Στ = -W*L*sin(theta) + mu*Nw*L*cos(theta) = 0

Now, we have four equations and
 

What is torque?

Torque is a measure of the force that can cause an object to rotate around an axis.

How is torque calculated?

Torque is calculated by multiplying the force applied by the distance from the axis of rotation.

What are some common units of torque?

The most common units of torque are Newton-meters (N*m) and foot-pounds (ft-lb).

What is the difference between torque and force?

Torque is a measure of the rotational force, while force is a measure of the overall strength or push/pull of an object.

How can I solve a torque problem?

To solve a torque problem, you will need to identify the given values, use the formula for torque, and solve for the unknown variable. It may also be helpful to draw a diagram and label all forces and distances involved.

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