A uniform cylinder of mass M sits on a fixed plane inclined at an angle theta.
A string is tied to the cylinder’s rightmost point, and a mass m hangs from the string,
as shown. Assume that the coefficient of static friction between the cylinder and the
plane is sufficiently large to prevent slipping. What is m, in terms of M and theta, if the
setup is in static equilibrium? Hint: use usual horizontal and vertical axes instead of
slope horizontal and vertical.

Question

A uniform cylinder of mass M sits on a fixed plane inclined at an angle theta. 
A string is tied to the cylinder’s rightmost point, and a mass m hangs from the string, 
as shown. Assume that the coefficient of static friction between the cylinder and the
plane is sufficiently large to prevent slipping. What is m, in terms of M and theta, if the 
setup is in static equilibrium? Hint: use usual horizontal and vertical axes instead of
slope horizontal and vertical.

Accepted Answer

∻Solution∻ ‖ a ⋮ Identify the forces acting on the cylinder: The gravitational force acting downwards is $$F_g = Mg$$, the normal force $$F_{normal}$$ acting perpendicular to the inclined plane, and the tension $$T$$ in the string acting horizontally. The mass $$m$$ hanging from the string exerts a downward force of $$mg$$. ‖ ‖ b ⋮ Apply the equilibrium conditions: For the cylinder to be in static equilibrium, the sum of forces in both the horizontal and vertical directions must be zero. In the horizontal direction, we have: $$T = F_{friction}$$, and in the vertical direction: $$F_{normal} = Mg - mg$$. ‖ ‖ c ⋮ Calculate the normal force: The normal force can be expressed as $$F_{normal} = Mg \cos( heta)$$, where $$ heta$$ is the angle of the incline. The frictional force can be expressed as $$F_{friction} = \mu_s F_{normal}$$, where $$\mu_s$$ is the coefficient of static friction. ‖ ‖ d ⋮ Set up the equations: From the vertical equilibrium, we have $$F_{normal} = Mg - mg$$. Substituting for $$F_{normal}$$ gives us: $$Mg \cos( heta) = Mg - mg$$. Rearranging this equation allows us to solve for $$m$$ in terms of $$M$$ and $$ heta$$. ‖ ‖ e ⋮ Solve for $$m$$: Rearranging the equation yields: $$mg = Mg - Mg \cos( heta)$$, which simplifies to $$m = M(1 - \cos( heta))$$. ‖ ∻Answer∻ ⚹ m = M(1 - \cos( heta}) ⚹ ∻Key Concept∻ ⚹ Static Equilibrium: For an object to be in static equilibrium, the net force and net torque acting on it must be zero. This involves balancing all forces acting in both horizontal and vertical directions. Equations: $$\sum F_x = 0$$ and $$\sum F_y = 0$$. ⚹ ∻Explanation∻ ⚹ By analyzing the forces acting on the cylinder and applying the conditions for static equilibrium, we can derive the relationship between the hanging mass $$m$$ and the mass of the cylinder $$M$$ in terms of the angle of inclination $$ heta$$. ⚹

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