a particle moves from rest at A on the surface of a smooth circular cy...
The given scenario involves a particle moving from rest at point A on the surface of a smooth circular cylinder with radius r. The particle then leaves the cylinder at point B. We need to determine the equation relating the angles alpha and beta.
Let's analyze the situation step by step:
1. Define the problem:
- A particle starts at rest at point A on the surface of a smooth circular cylinder.
- The particle leaves the cylinder at point B.
- We need to find the equation relating the angles alpha and beta.
2. Understand the geometry:
- The particle's motion is constrained to the surface of the cylinder, which means it moves in a circular path.
- The radius of the cylinder is denoted by r.
- The angles alpha and beta represent the positions of the particle on the cylinder's surface.
3. Derive the equation:
- To find the equation relating alpha and beta, we can consider the geometry of the situation.
- Let's assume that the particle travels a distance s along the surface of the cylinder.
- The arc length s can be expressed as s = r * alpha, where alpha is the angle in radians.
- Similarly, the arc length s' from point A to B can be expressed as s' = r * beta, where beta is the angle in radians.
- Since the particle starts at rest and leaves the cylinder at point B, we can equate these two arc lengths: r * alpha = r * beta.
- Simplifying this equation, we get alpha = beta.
4. Determine the correct equation:
- Comparing the derived equation alpha = beta with the given options, we find that the correct equation is 3 sin alpha = 2 cos beta (Option 1).
Therefore, the equation relating the angles alpha and beta is 3 sin alpha = 2 cos beta.
a particle moves from rest at A on the surface of a smooth circular cy...
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