A giant swing at a county fair consists of a vertical central shaft with a number of horizontal arms attahed at its upper end. Each arm supports a seat suspended from a 5.00 m long rod, the upper end of which is fastened to the arm at a point 3.00 m from the central shaft A. Find the time of one revolution of the swing if the rod is supporting the seat makes an angle of 30.0 degrees with the vertical. (b) does the angle depend on the weight of the passenger for a fiven rate of revolution.
When the shaft is turning and the rod is at a 30 degree angle to vertical, the distance of the seat from the axis of rotation is
R = 3 + 5 sin 30 = 5.50 meters.
The angular rotation speed w (in radians/s) is given by
M R w^2 = T sin 30
where T is the tension in the rod. Vertical equilibrium tells you that
M g = T cos 30
Take the ratio of those two equations to cancel out T and you get
Rw^2/g = tan 30
From w you can get the period, P.
P = 2 pi/w
Note that M cancels out. (That should help you answer part (b)
straight fax that wrong drwls
To find the time of one revolution of the swing, we need to consider the motion of the seat as it swings around the central shaft.
First, let's start by visualizing the problem. We have a central shaft, and attached to it are multiple horizontal arms. Each arm has a seat suspended from a 5.00 m long rod. The upper end of the rod is fastened to the arm at a point 3.00 m from the central shaft.
Now, when the swing is in motion, the seat and the rod form a right-angled triangle with the central shaft. The 5.00 m rod is the hypotenuse, and one of the other sides is the perpendicular distance from the seat to the central shaft, which is 3.00 m.
Since we know the lengths of two sides of the triangle, we can use trigonometry to find the length of the remaining side and then calculate the time of one revolution.
Let's call the remaining side x. Using trigonometry, we can say that:
sin(30.0 degrees) = x / 5.00 m
Rearranging the equation, we have:
x = 5.00 m * sin(30.0 degrees)
Now, we can calculate the value of x:
x = 5.00 m * 0.5
x = 2.50 m
This means that the perpendicular distance from the seat to the central shaft is 2.50 m.
Now, to find the time of one revolution, we can use the formula:
Time = Distance / Speed
In this case, the distance is the circumference of the circular path traveled by the seat, and the speed is the distance covered per unit time.
The circumference of a circle is given by the formula: Circumference = 2 * π * radius
In our case, the radius of the circular path is the perpendicular distance from the seat to the central shaft, which is 2.50 m.
So, the circumference of the circular path is:
Circumference = 2 * π * 2.50 m
Next, we need to determine the speed at which the seat travels. Remember that the swing is in motion, so the seat is constantly changing direction at the ends of the circular path. At the highest point, the speed is momentarily zero, and at the lowest point, the speed is highest.
However, since we are only interested in the time of one revolution, we can assume that the speed is constant throughout the entire motion. This assumption allows us to use the average speed for our calculation.
To find the average speed, we can divide the total distance traveled in one revolution (which is the circumference) by the time taken for one revolution.
Therefore, the average speed is:
Average Speed = Circumference / Time
Now, we can rearrange the formula to solve for time:
Time = Circumference / Average Speed
Substituting the values we already calculated:
Time = (2 * π * 2.50 m) / Average Speed
Now, the average speed depends on the rate of revolution. Let's assume that the rate of revolution is given as r revolutions per minute (rpm).
To convert r rpm to radians per second (rad/s), we multiply r by 2π/60 since there are 2π radians in one revolution and 60 seconds in one minute.
So, the average speed is:
Average Speed = r * 2π/60 rad/s
Now, substituting this into our time formula:
Time = (2 * π * 2.50 m) / (r * 2π/60 rad/s)
The 2π cancels out, and we are left with:
Time = (2.50 m) / (r / 60 rad/s)
Simplifying this further, we get:
Time = 150 m / r
Therefore, the time of one revolution is 150 m divided by the rate of revolution (r) in revolutions per minute (rpm).
(b) Now, let's address the question of whether the angle depends on the weight of the passenger for a given rate of revolution.
The angle in this problem is determined by the length of the rod and the perpendicular distance from the seat to the central shaft. It does not depend on the weight of the passenger.
The weight of the passenger affects the force exerted on the seat, which in turn affects the centripetal force required to keep the seat moving in a circular path. However, the angle remains the same regardless of the weight of the passenger.
So, in summary, the time of one revolution depends on the rate of revolution, and the angle does not depend on the weight of the passenger for a given rate of revolution.