To determine the minimum coefficient of static friction required for the hollow cylinder to roll without slipping on an inclined plane while containing a cube, we need to analyze the forces and torques acting on the system. Let's break this down step by step.
Understanding the System
We have a hollow cylinder with radius 'R' resting on an inclined plane at an angle of θ = 37°. Inside this cylinder, there is a cube with a face diagonal of 2R. The mass of both the cylinder and the cube is the same. The goal is to find the minimum coefficient of static friction (μ) between the cylinder and the inclined plane that allows the cylinder to roll without slipping.
Forces Acting on the Cylinder
- Weight (W): The weight of the cylinder acts downward and can be expressed as W = mg, where m is the mass and g is the acceleration due to gravity.
- Normal Force (N): The normal force acts perpendicular to the inclined surface.
- Frictional Force (f): This force acts parallel to the inclined plane and is responsible for the rolling motion.
Equations of Motion
For the cylinder to roll without slipping, the following conditions must be satisfied:
- The net force acting down the incline must equal the mass times the acceleration of the center of mass of the cylinder.
- The frictional force must provide the necessary torque for rolling.
Force Analysis
The forces acting on the cylinder can be resolved into components along the incline:
- The component of weight down the incline: W sin(θ) = mg sin(37°)
- The normal force: N = mg cos(θ) = mg cos(37°)
- The maximum static frictional force: f_max = μN = μmg cos(37°)
Net Force Equation
The net force acting on the cylinder along the incline can be expressed as:
mg sin(37°) - f = ma
Where 'a' is the linear acceleration of the cylinder's center of mass.
Torque and Rolling Condition
The torque (τ) about the center of mass due to friction is given by:
τ = fR
This torque must equal the moment of inertia (I) times the angular acceleration (α):
fR = Iα
For a hollow cylinder, the moment of inertia is I = mR². The relationship between linear acceleration (a) and angular acceleration (α) for rolling without slipping is:
a = Rα
Substituting α gives:
fR = mR²(a/R)
Thus, we have:
f = ma
Combining Equations
Now, we can substitute the expression for 'f' back into the net force equation:
mg sin(37°) - ma = μmg cos(37°)
Rearranging gives:
mg sin(37°) = ma + μmg cos(37°)
Factoring out 'm' yields:
g sin(37°) = a + μg cos(37°)
Finding Acceleration
From the torque equation, we also have:
a = \frac{f}{m}
Substituting this into our earlier equation gives:
g sin(37°) = \frac{f}{m} + μg cos(37°)
Now, substituting f = μmg cos(37°) into this equation leads to:
g sin(37°) = μg cos(37°) + μg cos(37°)
Which simplifies to:
g sin(37°) = 2μg cos(37°)
Solving for Coefficient of Friction
Dividing both sides by g and rearranging gives:
μ = \frac{sin(37°)}{2 cos(37°)}
Using the values of sin(37°) and cos(37°) (approximately 0.6 and 0.8 respectively), we find:
μ = \frac{0.6}{2 \times 0.8} = \frac{0.6}{1.6} = 0.375
Final Result
The minimum coefficient of static friction required for the hollow cylinder to roll without slipping on the inclined plane is approximately 0.375. This ensures that the forces and torques are balanced, allowing for pure rolling motion.
