To find the angle at which the sphere will slide and maintain a steady height, we need to equate the gravitational force with the centrifugal force acting on the sphere.
(a) Let's start with the first case where the period is 0.45 seconds.
The centripetal force acting on the sphere is given by the equation:
F_c = m * (2Ï€r / T)^2
Where:
F_c is the centripetal force,
m is the mass of the sphere,
r is the radius of the circular wire (15 cm = 0.15 m),
and T is the period of revolution (0.45 seconds).
The centripetal force is balanced by the gravitational force acting on the sphere:
F_g = m * g
Where:
F_g is the gravitational force,
m is the mass of the sphere,
and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Setting these two equations equal to each other, we have:
m * (2Ï€r / T)^2 = m * g
Simplifying the equation, we find the mass of the sphere cancels out:
(2Ï€r / T)^2 = g
Next, we can solve for the angle up from the bottom at which the sphere will slide.
Since the only force acting on the sphere is the gravitational force, we can equate the component of the gravitational force along the wire (mg * cosθ) with the centrifugal force (m * (2πr / T)^2) to find the angle θ:
mg * cosθ = m * (2πr / T)^2
Canceling out the mass, we get:
g * cosθ = (2πr / T)^2
Simplifying further, we can solve for cosθ:
cosθ = (2πr / T)^2 / g
For this specific case, substituting the given values, we get:
cosθ = (2π * 0.15 / 0.45)^2 / 9.8
Using a calculator, we can compute the value of cosθ and find the angle θ:
cosθ ≈ 0.4391
θ ≈ cos^(-1)(0.4391)
Therefore, the angle up from the bottom at which the sphere will slide and maintain a steady height is approximately θ ≈ 64.9621 degrees.
(b) Now let's consider the second case where the period is 0.85 seconds.
Following the same equation and steps as in part (a), we can calculate the angle θ for this case as well.
cosθ = (2πr / T)^2 / g
Substituting the given values:
cosθ = (2π * 0.15 / 0.85)^2 / 9.8
Using a calculator, we can compute the value of cosθ and find the angle θ:
cosθ ≈ 0.2659
θ ≈ cos^(-1)(0.2659)
Therefore, the angle up from the bottom at which the sphere will slide and maintain a steady height is approximately θ ≈ 75.0521 degrees.