Problem: Find the angle between the cord and the vertical axis when a heavy ball is suspended from a fixed point by a string of length 1m and is rotating about a vertical axis through this point with uniform angular velocity of 10 rad/sec.

Solution:

Step 1: Understand the problem
The problem involves a heavy ball that is suspended from a fixed point by a string of length 1m. The ball is rotating about a vertical axis through this point with a uniform angular velocity of 10 rad/sec. We need to find the angle between the cord and the vertical axis.

Step 2: Draw a diagram
A diagram can help us understand the problem better and visualize the situation.



Step 3: Analyze the problem
We know that the ball is rotating about a vertical axis with a uniform angular velocity of 10 rad/sec. This means that the angular displacement of the ball is 10 rad/sec for every second of time. We also know that the length of the string is 1m.

We can use trigonometry to find the angle between the cord and the vertical axis. Let's assume that the angle between the string and the vertical axis is θ. From the diagram, we can see that:

cos(θ) = adjacent/hypotenuse

cos(θ) = r/1

cos(θ) = r

where r is the radius of the circle traced by the ball.

We can also use the formula for angular velocity:

ω = Δθ/Δt

where ω is the angular velocity, Δθ is the angular displacement, and Δt is the time interval.

If we rearrange this formula, we get:

Δθ = ω x Δt

Substituting the values given in the problem, we get:

Δθ = 10 rad/sec x 1 sec

Δθ = 10 rad

This means that the ball has rotated 10 radians in 1 second.

Step 4: Solve the problem
Using the formula cos(θ) = r, we can find the value of cos(θ):

cos(θ) = r

cos(θ) = 1

θ = cos^-1(1)

θ = 0 degrees

Therefore, the angle between the cord and the vertical axis is 0 degrees.

Step 5: Check the solution
We can check the solution by verifying that the angle between the cord and the vertical axis is indeed 0 degrees. From the diagram, we can see that the cord is vertical, which means that the angle between the cord and the vertical axis is 0 degrees.

Conclusion: In conclusion, the angle between the cord and the vertical axis is 0 degrees.