Due to friction ,the system as shown in the diagram remains motionless.
Calculate the static coefficient of friction ?

Question

William Audy · Accepted Answer

#mu_s<=0.346#

What I tried is this.

Forces on #m_1#:

#sumF_x=T_1-(F_G)_x-f_s=0#

#sumF_y=n-(F_G)_y=0#

#f_(s"max")=mu_sn#

#(F_G)_x=m_1gsin(theta)#

#(F_G)_y=m_2gcos(theta)#

#n=mgcos(theta)#

I will refer to #f_(s"max")# simply as #f_s# from this point on, though I am still solving in terms of the maximum static friction.

#=>f_s=T_1-m_1gsin(theta)#

#=>mu_sm_1gcos(theta)=T_1-m_1gsin(theta)#

#=>color(darkblue)(mu_s=(T_1-m_1gsin(theta))/(m_1gcos(theta)))#

Forces on #m_2#:

#sumF=sumF_y=T_2-F_G=0#

Because we can assume a massless rope and frictionless pulley, #vecT_1# and #vecT_2# act as an "action/reaction" pair.

#=>mu_s=(m_2g-m_1gsin(theta))/(m_1gcos(theta))#

#=>mu_s=(cancel(g)(m_2-m_1sin(theta)))/(cancel(g)(m_1cos(theta))#

#=>color(darkblue)(mu_s=(m_2-m_1sin(theta))/(m_1cos(theta)))#

Applying recognized values

#mu_s=(80-100(0.500))/(100(0.866))#

#mu_(s"max")=0.346#

#=>mu_s<=0.346#

Nicholas Agrusa · Answer

undefined To calculate the static coefficient of friction, we use the formula:

Static friction force (Fs) = Static coefficient of friction (μs) × Normal force (N)

Given that the system remains motionless, the static friction force balances the applied force.

First, we need to determine the normal force (N), which is the force exerted perpendicular to the surface.

Next, we can use the equation for equilibrium in the vertical direction:

N - mg = 0

Where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Once we have the normal force (N), we can use the equation for static friction force:

Fs = μs × N

Since the system remains motionless, the static friction force equals the applied force, which is the weight of the object (mg).

Therefore:

μs × N = mg

And solving for μs:

μs = mg / N

Substitute the expression for N from the equilibrium equation:

μs = mg / (mg)

μs = 1

So, the static coefficient of friction is 1.