To determine the minimum radius of the semicircular cylinder that allows the wooden bar to be in stable equilibrium, we need to analyze the balance of forces and moments acting on the system. The key concept here is the center of mass and how it relates to the support point provided by the semicircular cylinder.
Understanding the System
The wooden bar is homogeneous, meaning its mass is evenly distributed. The dimensions given are:
- Length of the bar (L) = 10 cm
- Thickness of the bar (T) = 4 cm
- Weight of the bar (W) = 1 kg
To find the minimum radius (R) of the semicircular cylinder, we need to consider the position of the center of mass of the bar and how it relates to the point of contact with the cylinder.
Finding the Center of Mass
The center of mass of the wooden bar is located at its midpoint, which is at a distance of 5 cm from either end. When the bar is placed on the semicircular cylinder, the stability depends on whether the center of mass remains directly above the point of contact with the cylinder.
Geometric Considerations
For the bar to be in stable equilibrium, the center of mass must not extend beyond the edge of the semicircular cylinder. The semicircular cylinder can be visualized as having a radius R, and the center of mass of the bar must be positioned such that it does not fall outside this radius.
Calculating the Minimum Radius
When the bar is balanced on the semicircular cylinder, the distance from the center of mass to the edge of the cylinder can be expressed in terms of the radius R. The critical condition for stability occurs when the center of mass is directly above the edge of the semicircle.
To maintain stability, the following relationship must hold:
- Distance from the center of the semicircle to the center of mass = R
- Height of the center of mass above the base = 4 cm (the thickness of the bar)
Using the Pythagorean theorem, we can relate these distances. The horizontal distance from the center of the semicircle to the center of mass is half the length of the bar:
- Horizontal distance = 5 cm
Thus, we can set up the equation:
R^2 = (5 cm)^2 + (4 cm)^2
Calculating this gives:
- R^2 = 25 cm² + 16 cm²
- R^2 = 41 cm²
- R = √41 cm ≈ 6.4 cm
Final Result
The minimum radius of the semicircular cylinder required for the wooden bar to be in stable equilibrium is approximately 6.4 cm. This ensures that the center of mass of the bar remains directly above the point of contact, preventing it from tipping over.
