A mass M can slide with negligible friction inside a tube bent in a circle of radius R = 1.45 m. The tube rotates about its diameter; the mass is in equilibrium at ¦È = 47.3¡ã. Calculate the number of revolutions the hoop makes in 115 s.

To calculate the number of revolutions the hoop makes in 115 seconds, we need to determine the time it takes for one revolution and then divide the total time by that period.

The time it takes for one revolution can be found using the concept of angular velocity. Angular velocity (ω) is defined as the change in angle (θ) per unit of time (t). In this case, the hoop rotates with a constant angular velocity.

To find the angular velocity, we can use the equation: ω = Δθ / Δt.

Given that the hoop rotates in equilibrium at an angle of θ = 47.3 degrees and we want to find the time for one revolution, we should note that one revolution corresponds to a change in angle of 360 degrees (or 2π radians).

Converting the angle to radians: θ = 47.3 degrees * (π / 180 degrees) = 0.826 radians.

Substituting these values into the equation, we have: ω = 0.826 radians / Δt.

Rearranging the equation to solve for Δt, we get: Δt = 0.826 radians / ω.

Now, we need to determine the angular velocity from the given information. The mass is in equilibrium, which means it experiences a centripetal force that balances the gravitational force. The centripetal force (Fc) is given by the equation: Fc = M * R * ω^2, where M is the mass, R is the radius of the circle, and ω is the angular velocity.

Since there is no friction, the only force acting on the mass is the gravitational force (Fg = Mg), where g is the acceleration due to gravity.

Setting Fc equal to Fg, we get: M * R * ω^2 = Mg.

Simplifying the equation by canceling out M and rearranging, we have: ω^2 = g / R.

Now, we can substitute the given values into the equation to find the angular velocity.

The radius of the circle, R, is given as 1.45 m, and the acceleration due to gravity, g, is approximately 9.8 m/s^2.

Substituting these values, we get: ω^2 = 9.8 m/s^2 / 1.45 m = 6.7586 rad/s^2.

Taking the square root of both sides to find ω, we get: ω = √(6.7586 rad/s^2) ≈ 2.6 rad/s.

Now that we have the angular velocity, we can substitute it back into the equation Δt = 0.826 radians / ω to find the time for one revolution.

Δt = 0.826 radians / 2.6 rad/s ≈ 0.318 s.

Finally, to calculate the number of revolutions in 115 seconds, we divide the total time by the time for one revolution.

Number of revolutions = 115 s / 0.318 s ≈ 361.32 revolutions.

Therefore, the hoop makes approximately 361.32 revolutions in 115 seconds.

To solve this problem, we need to use the concept of centripetal force.

1. Given:
- Mass (M) of the object = ?
- Radius (R) of the circle = 1.45 m
- Angle (θ) = 47.3°
- Time (t) = 115 s

2. First, let's find the centripetal force acting on the mass:

Centripetal force (F) is given by the equation:
F = M * ω^2 * R

Where:
- F = centripetal force
- M = mass
- ω = angular velocity
- R = radius

3. We need to find the angular velocity ω. The angular velocity is given by the equation:

ω = θ / t

Where:
- θ = angle in radians
- t = time in seconds

Convert the angle from degrees to radians:
θ = (47.3 * π) / 180

4. Plug in the values into the equation to find angular velocity:

ω = (θ / t)
ω = ((47.3 * π) / 180) / 115

5. Now, we can calculate the centripetal force:

F = M * ω^2 * R

Rearranging the equation to solve for M:
M = F / (ω^2 * R)

6. Next, calculate the number of revolutions (N) the hoop makes in 115 s:

Number of revolutions (N) = ω * t / (2π)

Plug in the values to calculate the number of revolutions:
N = (ω * t) / (2π)

Now, let's plug in the values and calculate step-by-step.