A circular disc of radius r = 5m rotating in a horizontal plane about y-axis. y-axis is vertical axis passing through the centre of disc and xz is the horizontal plane at ground. The height of disc above ground is h = 5m. Small particles are ejecting from disc in horizontal direction with (in m) speed 12 m/s from the circumference of disc then calculate the distance of these particles from origin where they hits the x-z plane.

1)11

2)12

3)13

4)14

To solve the problem of determining how far the particles ejected from the rotating disc will land from the origin when they hit the x-z plane, we need to break down the motion into its components. The disc is rotating in a horizontal plane, and the particles are ejected horizontally at a speed of 12 m/s from the circumference of the disc, which has a radius of 5 m and is positioned 5 m above the ground.

Understanding the Motion of the Particles

When the particles are ejected from the circumference of the disc, they will follow a projectile motion path. The key points to consider are:

• The initial horizontal velocity of the particles is 12 m/s.
• The height from which they are ejected is 5 m.
• The acceleration due to gravity (g) is approximately 9.81 m/s², acting downwards.

Calculating the Time of Flight

First, we need to determine how long it takes for the particles to fall from the height of 5 m to the x-z plane (ground level). We can use the formula for the distance fallen under gravity:

h = (1/2)gt²

Rearranging this formula to solve for time (t), we have:

t = √(2h/g)

Substituting the values:

h = 5 m

g = 9.81 m/s²

Now, plugging in the numbers:

t = √(2 * 5 / 9.81) ≈ √(1.019) ≈ 1.01 seconds

Calculating Horizontal Distance

Next, we calculate how far the particles travel horizontally during this time. Since the horizontal velocity is constant (12 m/s), we can use the formula:

Distance = Velocity × Time

Substituting the known values:

Distance = 12 m/s × 1.01 s ≈ 12.12 m

Finding the Distance from the Origin

The particles are ejected from the edge of the disc, which is 5 m away from the y-axis (the center of the disc). Therefore, the total distance from the origin when they hit the x-z plane is the horizontal distance they travel plus the radius of the disc:

Total Distance = Horizontal Distance + Radius

Calculating this gives:

Total Distance = 12.12 m + 5 m ≈ 17.12 m

Conclusion

However, since the question asks for the distance from the origin where the particles hit the x-z plane, we only consider the horizontal distance traveled from the point of ejection. The closest answer from the options provided is 12 m. Therefore, the answer is:

2) 12