A stick of length l =70.0cm rests against the wall. The coefficient of static friction between stick and wall and between the stick and the floor are equal. The stick will slip off the wall if placed at an angle greater than theta = 41 degrees. What is the the coefficient of static friction between the stick and the wall and the floor? Thanks.
tan (theta)= (2*u_s)/(1-u_s^2)
,solve for u_s
Hi, koala. I am having trouble solving for u_s. So far,
tan(theta) = 2*(u_s)/(1-u_s)*(u_s+1)
u_s=?
In this problem, we can use the concept of equilibrium to find the coefficient of static friction between the stick and the wall and the floor.
Let's start by analyzing the forces acting on the stick when it is placed against the wall. There are two main forces involved: the weight of the stick acting downwards and the friction force acting in the opposite direction (upwards) to prevent the stick from sliding down.
To find the coefficient of static friction, we need to find the maximum angle at which the stick is about to slip off the wall. At this point, the friction force reaches its maximum value before the stick starts to slide.
First, let's consider the forces acting on the stick perpendicular to the wall. We can resolve the weight of the stick into two components: one perpendicular to the wall (mg * cos(theta)) and one parallel to the wall (mg * sin(theta)). The friction force acts in the opposite direction of the component of the weight parallel to the wall.
Now, the stick will be in equilibrium along the perpendicular direction (away from the wall). This means that the sum of the forces along that direction is zero. Therefore, we can write the following equation:
mg * cos(theta) - friction_force = 0
We can rearrange this equation to solve for the friction force:
friction_force = mg * cos(theta)
Next, let's consider the forces acting on the stick parallel to the wall. The friction force is acting towards the wall, and the component of the weight parallel to the wall is acting away from the wall. Thus, the equation for this direction becomes:
friction_force - mg * sin(theta) = 0
Now that we have expressions for the friction force, we can equate them and solve for the coefficient of static friction, denoted as μ:
mg * cos(theta) = mg * sin(theta)
Dividing both sides of the equation by mg, we get:
cos(theta) = sin(theta)
Since the coefficient of static friction is the same for both surfaces, we can replace sin(theta) with cos(theta) in the equation:
cos(theta) = cos(theta)
This means that the coefficient of static friction between the stick and the wall and between the stick and the floor are equal.
Therefore, the coefficient of static friction between the stick and the wall and the floor is equal to 1 in this case.