Understanding the Physics

When you swing a bucket of water in a vertical circle, the water experiences two forces:

* Gravity: Pulling the water downwards.

* Centripetal force: The force that keeps the water moving in a circular path.

To prevent spilling, the centripetal force must be *greater than or equal to* the force of gravity at the top of the circle.

Setting up the Equations

* Centripetal force (Fc): Fc = m * v^2 / r, where:

* m is the mass of the water

* v is the linear velocity of the water

* r is the radius of the circle (the length of the student's arm)

* Force of gravity (Fg): Fg = m * g, where:

* g is the acceleration due to gravity (approximately 9.8 m/s^2)

Solving for Angular Velocity

1. Equal Forces: At the top of the circle, Fc = Fg to prevent spilling.

m * v^2 / r = m * g

2. Canceling mass: The mass of the water cancels out:

v^2 / r = g

3. Linear velocity to angular velocity: We need to relate linear velocity (v) to angular velocity (ω). The relationship is: v = r * ω

4. Substituting and solving for ω:

(r * ω)^2 / r = g

r * ω^2 = g

ω^2 = g / r

ω = √(g / r)

Calculating the Result

* Convert arm length to meters: 63 cm = 0.63 m

* Plug in values: ω = √(9.8 m/s^2 / 0.63 m) ≈ 3.95 rad/s

Answer: The minimum angular velocity required to prevent spilling is approximately 3.95 rad/s.