A girl on a merry-go-round platform holds a pendulum in her hand. The pendulum is 3.7 m from the rotation axis of the platform. The rotational speed of the platform is 0.025 rev/s. It is found that the pendulum hangs at an angle è to the vertical. Find è.
The angular speed of the platform is
w = (0.025 rev/s)*(2 pi) rad/rev
= 0.157 rad/s
The centripetal acceleration of the pendulum is
a_c = R*w^2 = 0.0913 m/s^2
The angle theta is
tan^-1(a_c/g) = 0.918 degrees
a girl on a rotating platform holds a pendulum in her hand. The pendulum is
at a radius of 6.0 m from the center of the platform. The rotational speed of the platform is 0.020 rev/s.
It is found that the pendulum hangs at an angle θ to the vertical, as shown. Find θ.
To find the angle (θ) at which the pendulum hangs, we can use the concept of the centripetal force acting on the pendulum.
The centripetal force acting on the pendulum can be calculated using the following formula:
Fc = m * ω^2 * r
Where:
- Fc is the centripetal force
- m is the mass of the object (assuming it's concentrated at the end of the pendulum)
- ω is the angular velocity of the merry-go-round platform
- r is the distance between the rotation axis and the pendulum
In this case, we are given that the rotational speed of the platform is 0.025 rev/s, and the distance between the rotation axis and the pendulum is 3.7 m.
Since we are only interested in the angle (θ) at which the pendulum hangs, we need to find the ratio between the gravitational force (mg) and the centripetal force (Fc).
The gravitational force acting on the pendulum is given by:
Fg = mg
Assuming there is no other force acting on the pendulum besides gravity and centripetal force, these two forces must be equal in magnitude:
Fc = Fg
Substituting the formulas for Fc and Fg, we have:
m * ω^2 * r = mg
Canceling out the mass (m) from both sides of the equation, we get:
ω^2 * r = g
Rearranging the equation to solve for ω:
ω = √(g/r)
Substituting the given values for g (acceleration due to gravity) as 9.8 m/s^2 and r as 3.7 m, we can calculate ω:
ω = √(9.8/3.7) = √(2.648) ≈ 1.63 rad/s
Finally, to find the angle (θ), we can use the formula for the angle of a pendulum:
θ = sin^(-1)(ωt/g)
where:
- t is the time taken for one full revolution of the platform, which can be calculated as 1/ω (since ω is the angular velocity in rad/s)
Substituting the values, we have:
θ = sin^(-1)(1.63 * (1/0.025) / 9.8)
Evaluating this expression, we find:
θ ≈ sin^(-1)(65.2 / 9.8) ≈ sin^(-1)(6.651) ≈ 74.3 degrees
Therefore, the pendulum hangs at an angle of approximately 74.3 degrees to the vertical.
To find the angle è, you need to consider the forces acting on the pendulum and use the concept of centripetal force. Here's a step-by-step explanation of how to find the angle è:
1. First, let's assume that the girl and the merry-go-round are in a stable state, meaning the centripetal force acting on the pendulum equals the gravitational force acting on it.
2. The centripetal force is the force that keeps an object moving in a circle. In this case, the centripetal force is provided by the tension in the string of the pendulum.
3. The gravitational force is given by the weight of the pendulum, which can be calculated as the mass of the pendulum times the acceleration due to gravity (F = m * g).
4. The tension in the string can be expressed as the equation: T = m * a, where T is the tension, m is the mass of the pendulum, and a is the centripetal acceleration.
5. The centripetal acceleration can be calculated using the formula: a = r * ω^2, where a is the centripetal acceleration, r is the radius of the circular path (in this case, the length of the pendulum), and ω is the angular velocity of the platform.
6. Convert the angular velocity from rev/s to rad/s by multiplying it by 2π since 1 revolution is equal to 2π radians (ω = 0.025 rev/s * 2π rad/rev).
7. Plug in the values you have into the equations. The mass of the pendulum cancels out, and you're left with: T = r * ω^2 = m * g.
8. Solve for the angle è using the equation: tan(è) = (T / m * g). Rearrange the equation to find è: è = arctan(T / m * g).
Now, you have all the information and equations necessary to find the angle è. Plug in the values for the radius of the pendulum, the angular velocity of the platform, the mass of the pendulum, and the acceleration due to gravity to calculate the angle è.