A doubly tethered model airplane of mass 1.0 kg is attached to a vertical rod by two very light strings that are 2.0 each in length. the airplane and the the strings rotate in a horizontal circleabout the rod with both the strings remaining taut.The attachment points of the strings are also separated by 2.0 m If the tension in the upper string is 39.2 N. Calculate
a)the tension in the lower string.Is it likely to be less or more than the tension in the upper string?
b)the steady speed of the ball as it undergoes circular motion.
If the tensions were unequal, wouldn't there be a bending moment causing the model to twist?
The speed of the plane (not ball) causes centripetal force, which has to equal the sum of tensions.
a) To calculate the tension in the lower string, we can use the concept of equilibrium in circular motion. In a system where an object is moving in a circular motion at a constant speed, the net force acting on the object should be directed towards the center of the circle. This net force is provided by the tension in the strings.
In this case, since the strings are taut and the object is rotating in a horizontal circle, the vertical component of tension in the upper string balances the weight of the object, and therefore there is no vertical acceleration.
First, let's find the vertical component of tension in the upper string:
T_vertical = T_upper * cos(theta)
where T_upper is the tension in the upper string and theta is the angle formed between the upper string and the vertical.
To find theta, we can use the given information that the strings are 2.0 m in length and their attachment points are separated by 2.0 m. This forms a right-angled triangle, and theta is the angle opposite the side of length 2.0 m. Using trigonometry, we can find theta as follows:
tan(theta) = opposite/adjacent
tan(theta) = (2.0 m) / (2.0 m)
theta = arctan(1.0) = 45 degrees
Now, we can calculate the vertical component of tension:
T_vertical = 39.2 N * cos(45 degrees) = 39.2 N * 0.707 = 27.7 N
Since the system is in equilibrium, the net force in the vertical direction is zero, meaning the sum of the vertical component of tension in the lower string and the weight of the object is zero. Therefore, the tension in the lower string must balance the weight of the object:
T_lower * cos(theta) = weight
T_lower * cos(45 degrees) = (mass * gravity)
T_lower = (1.0 kg * 9.8 m/s^2) / cos(45 degrees)
T_lower = 9.8 N / 0.707
T_lower = 13.9 N
So, the tension in the lower string is 13.9 N. It is less than the tension in the upper string.
b) The steady speed of the model airplane can be calculated using the concept of centripetal force. In circular motion, the centripetal force is provided by the tension in the strings.
The net force acting towards the center of the circular path is given by:
Fc = T_upper + T_lower
Since the tension in the lower string is less than the tension in the upper string, the net force towards the center of the circle will be smaller than the tension in the upper string.
Now, we can equate the centripetal force to the required force for circular motion, which is the mass times the centripetal acceleration:
Fc = mass * (velocity^2 / radius)
Substituting the known values:
(39.2 N + 13.9 N) = 1.0 kg * (velocity^2 / 2.0 m)
Simplifying the equation:
53.1 N = 0.5 kg * velocity^2
velocity^2 = (53.1 N / 0.5 kg)
velocity^2 = 106.2 m^2/s^2
Taking the square root of both sides:
velocity = sqrt(106.2 m^2/s^2)
velocity ≈ 10.3 m/s
Therefore, the steady speed of the model airplane as it undergoes circular motion is approximately 10.3 m/s.