A person is flying a model airplane with mass 82.2 gram in a horizontal circular path on the
end of a string 10.5 m long. The string is horizontal and exerts a force of 3.22 N on the hand
of the person holding it. What is the speed (in m/s) of the plane?
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To find the speed of the model airplane, we can use the concept of centripetal force. Centripetal force is the force that keeps an object moving in a circular path.
The centripetal force in this case is provided by the tension in the string, which is the force exerted by the string on the airplane. In order to use this information, we need to calculate the tension in the string.
The tension in the string can be calculated using Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. Since the airplane is moving in a circular path, its acceleration is towards the center of the circle.
First, convert the mass of the airplane to kilograms:
mass = 82.2 grams = 82.2 / 1000 = 0.0822 kg
Next, we can calculate the tension in the string using Newton's second law:
Tension = mass * acceleration
The acceleration of the airplane is given by the centripetal acceleration formula:
acceleration = (velocity^2) / radius
The radius of the circular path is given as 10.5 m.
Now, let's substitute the known values into the equation to solve for tension:
Tension = mass * ((velocity^2) / radius)
We also know that the tension in the string is 3.22 N:
3.22 N = (0.0822 kg) * ((velocity^2) / 10.5 m)
Now, we can solve for the velocity:
velocity^2 = (3.22 N * 10.5 m) / 0.0822 kg
Take the square root on both sides to find the velocity:
velocity = sqrt((3.22 N * 10.5 m) / 0.0822 kg)
Now, let's plug in the values and calculate the velocity:
velocity = sqrt((3.22 * 10.5) / 0.0822) m/s
Simplifying the expression gives:
velocity = sqrt(416.1) m/s
Finally, calculating the square root gives:
velocity ≈ 20.4 m/s
Therefore, the speed of the model airplane is approximately 20.4 m/s.