A bar of mass m lies on a table attached to a stationary point O, with a non deformed elastic thread of length lo. The table was moved slowly on the floor till the bar started to slip when the thread made an angle theta with the vertical. If the coefficient of friction between bar and floor is k, find the work done by static friction acting on the bar till it started to slip, in the reference frame attached to the floor.

To tackle this problem, we need to analyze the forces acting on the bar as it begins to slip due to the movement of the table. The key here is to understand the role of static friction and how it interacts with the forces exerted by the elastic thread and the weight of the bar.

Understanding the Forces at Play

Initially, when the table is moved, the bar remains stationary due to static friction. The static friction force acts to prevent the bar from slipping. As the table continues to move, the elastic thread exerts a force that can be broken down into two components: one that acts vertically and another that acts horizontally. The angle θ that the thread makes with the vertical is crucial in determining these components.

Breaking Down the Forces

Let’s denote the following:

• m: mass of the bar
• g: acceleration due to gravity
• k: coefficient of static friction
• l₀: original length of the elastic thread
• F_s: force of static friction
• T: tension in the elastic thread

The weight of the bar acts downward with a force of mg. The tension T in the elastic thread can be expressed as:

T = mg * cos(θ)

The horizontal component of the tension, which contributes to the slipping of the bar, is:

T_x = T * sin(θ) = mg * cos(θ) * sin(θ)

Static friction must balance this horizontal component until the bar begins to slip. The maximum static friction force can be calculated as:

F_s,max = k * N = k * mg

Condition for Slipping

The bar will start to slip when the horizontal component of the tension exceeds the maximum static friction force:

mg * cos(θ) * sin(θ) > k * mg

Dividing through by mg (assuming m is not zero), we get:

cos(θ) * sin(θ) > k

Calculating Work Done by Static Friction

Once the bar starts to slip, the work done by static friction can be calculated. The work done by a force is given by:

W = F * d * cos(φ)

In this case, the static friction force acts in the opposite direction to the displacement of the bar. Therefore, φ = 180°, and cos(180°) = -1. The distance d is the distance the bar moves before slipping begins.

Thus, the work done by static friction is:

W_f = -F_s,max * d

Substituting for F_s,max:

W_f = -k * mg * d

Final Thoughts

In summary, the work done by static friction acting on the bar until it starts to slip is given by:

W_f = -k * mg * d

This negative sign indicates that the work done by static friction is in the direction opposite to the displacement of the bar. Understanding these forces and their interactions is crucial in solving problems related to friction and motion.