if two inclines are placed such that the left one is having pointed end towards right and the right one having pointed end towards right and then dey are joined such that pointed of one touches heigtened of other, then find the velocity of a cylinder, if left incline moves with u m/s towards left and right one with 2u towards right. incines are at 30 degree from horizontal, and cylinder remains in contact thrghout the process?

To solve this problem, we need to analyze the motion of the cylinder as it rolls down the two inclined planes. The setup involves two inclines, both at an angle of 30 degrees to the horizontal, with one incline moving to the left at a speed of u m/s and the other moving to the right at a speed of 2u m/s. The key is to understand how the motion of the inclines affects the velocity of the cylinder.

Understanding the System

First, let's visualize the scenario. We have two inclined planes:

• The left incline moves leftward with a speed of u m/s.
• The right incline moves rightward with a speed of 2u m/s.

When these inclines are joined at their pointed ends, the cylinder rolls down the left incline and then onto the right incline. The cylinder remains in contact with both inclines throughout the motion.

Analyzing the Motion

As the cylinder rolls down the left incline, it experiences gravitational acceleration. The component of gravitational force acting along the incline can be calculated using the angle of the incline:

• Gravitational force component along the incline = mg sin(30°) = (1/2)mg.

However, since the incline is moving, we need to consider the relative motion of the cylinder with respect to the inclines.

Relative Velocity Considerations

When the cylinder is on the left incline, it rolls down while the incline itself moves left. The effective velocity of the cylinder relative to the ground can be expressed as:

• Velocity of the cylinder (v) = velocity of incline (u) + velocity due to gravity along the incline.

As the cylinder rolls down, it gains speed due to the gravitational pull. The acceleration of the cylinder down the incline can be expressed as:

• Acceleration (a) = g sin(30°) = (1/2)g.

Velocity Calculation

Now, let's consider the motion on the right incline. The right incline moves to the right at 2u m/s. When the cylinder reaches the top of the right incline, its velocity will be influenced by the motion of the incline:

• Effective velocity on the right incline = velocity of cylinder + velocity of incline = v + 2u.

To find the final velocity of the cylinder, we can apply the principle of conservation of energy or analyze the forces acting on the cylinder. However, since the inclines are moving, we can also use the relative motion approach.

Final Velocity Expression

Assuming the cylinder rolls without slipping, the final velocity of the cylinder can be derived from the relative velocities:

• Let the final velocity of the cylinder be v.
• On the left incline, the effective velocity is v - u.
• On the right incline, the effective velocity is v + 2u.

Setting these equal gives us a relationship to solve for v:

• v - u = v + 2u.

Solving this equation leads to:

• v - u - v - 2u = 0,
• -3u = 0,
• v = 3u.

Conclusion

The final velocity of the cylinder, as it rolls down the two inclined planes, is 3u m/s. This result shows how the motion of the inclines influences the overall speed of the cylinder, demonstrating the principles of relative motion and energy conservation in a dynamic system.