Problem: 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 10rad/s. Find the angle between cord and vertical axis.

Solution:

Step 1: Understanding the problem statement

The problem statement describes a system where a heavy ball is suspended from a fixed point by a string of length 1m. The system is rotating about a vertical axis through this point with uniform angular velocity of 10rad/s. We need to find the angle between cord and vertical axis.

Step 2: Identifying the relevant formulae

We can use the formula for centripetal force to solve this problem. The formula is given as:

F = mv²/r

where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.

We can also use the formula for gravitational force to calculate the weight of the object. The formula is given as:

Fg = mg

where Fg is the weight of the object, m is the mass of the object, and g is the acceleration due to gravity.

Step 3: Solving the problem

The weight of the ball can be calculated using the formula for gravitational force:

Fg = mg

where m is the mass of the ball and g is the acceleration due to gravity. Since the ball is hanging vertically, the weight is acting downwards. Therefore, the weight can be resolved into two components: one acting in the direction of the string and the other acting perpendicular to the string.

The component of weight acting in the direction of the string is given by:

F// = Fg sin θ

where θ is the angle between the string and the vertical axis.

The component of weight acting perpendicular to the string is given by:

F⊥ = Fg cos θ

Since the ball is rotating with uniform angular velocity, the centripetal force acting on the ball is given by:

F = mv²/r

where m is the mass of the ball, v is the velocity of the ball, and r is the radius of the circular path.

We can equate the two forces to get:

F// = F

Fg sin θ = mv²/r

Also, we can use the Pythagorean theorem to get:

r² = (1 - F⊥/Fg)²

Substituting the value of F⊥ from above, we get:

r² = (1 - cos θ)²

Solving for cos θ, we get:

cos θ = 1 - r²

__________

1

Substituting the values of r and v, we get:

cos θ = 1 - (v²/g)²

____________

1

Putting the given values, we get:

cos θ = 0.981

Therefore, the angle between the cord and vertical axis is:

θ = cos^-1 (0.981) = 12.5°

Step 4: Conclusion

The angle between the cord and vertical axis is 12.5°.