A model airplane with a mass of 0.748 kg is tethered by a wire so that it flies in a circle 30.2 m in radius. The airplane engine provides a net thrust of 0.806 N perpendicular to the tethering wire.
(a) Find the torque the net thrust produces about the center of the circle.
N·m
(b) Find the angular acceleration of the airplane when it is in level flight.
rad/s2
(c) Find the linear acceleration of the airplane tangent to its flight path.
m/s2
torque= force*radius
torque= mass*radius^2 * angular acceleration <--- NOT torque * radius
linear acceleraltion=angularacceleration*radius
(a) To find the torque produced by the net thrust about the center of the circle, we can use the formula:
Torque = Force * Radius
Given that the net thrust force is 0.806 N and the radius of the circle is 30.2 m, we can substitute these values into the formula:
Torque = 0.806 N * 30.2 m
Calculating the value:
Torque = 24.3712 N·m
Therefore, the torque produced by the net thrust is 24.3712 N·m.
(b) To find the angular acceleration of the airplane when it is in level flight, we can use the formula:
Torque = Moment of Inertia * Angular Acceleration
The moment of inertia for a point mass rotating around a fixed axis is given by:
Moment of Inertia = Mass * Radius^2
Given that the mass of the airplane is 0.748 kg and the radius of the circle is 30.2 m, we can calculate the moment of inertia:
Moment of Inertia = 0.748 kg * (30.2 m)^2
Calculating the value:
Moment of Inertia = 657.3096 kg·m^2
Now, we can rearrange the formula to solve for the angular acceleration:
Angular Acceleration = Torque / Moment of Inertia
Substituting the values:
Angular Acceleration = 24.3712 N·m / 657.3096 kg·m^2
Calculating the value:
Angular Acceleration = 0.0371 rad/s^2
Therefore, the angular acceleration of the airplane when it is in level flight is 0.0371 rad/s^2.
(c) The linear acceleration of the airplane tangent to its flight path can be found using the formula:
Linear acceleration = Angular acceleration * Radius
Given that the angular acceleration is 0.0371 rad/s^2 and the radius of the circle is 30.2 m, we can substitute these values into the formula:
Linear acceleration = 0.0371 rad/s^2 * 30.2 m
Calculating the value:
Linear acceleration = 1.12042 m/s^2
Therefore, the linear acceleration of the airplane tangent to its flight path is 1.12042 m/s^2.
To answer these questions, we need to apply some basic principles of physics, specifically related to rotational motion and circular motion.
(a) Torque is calculated as the product of the distance from the axis of rotation (r) and the force applied perpendicular to that distance (F):
Torque = r * F * sin(θ)
In this case, the distance from the center of the circle to the airplane (r) is the radius of the circle, which is given as 30.2 m. The net thrust of the engine (F) is given as 0.806 N. Since the airplane is flying perpendicular to the tethering wire, the angle (θ) between the radius and the force is 90 degrees, so sin(θ) is equal to 1.
Torque = 30.2 m * 0.806 N * 1 = 24.4012 N·m (rounding to four decimal places)
Therefore, the torque the net thrust produces about the center of the circle is approximately 24.4012 N·m.
(b) The torque produced by the net thrust causes the airplane to experience a rotational motion. The angular acceleration (α) can be determined using the following equation:
Torque = moment of inertia * angular acceleration
Given that the torque is 24.4012 N·m and the moment of inertia (I) is dependent on the shape and mass distribution of the airplane, we need more information to directly compute the angular acceleration.
(c) To find the linear acceleration of the airplane tangent to its flight path, we need to use the relation between linear and angular acceleration, given by the formula:
Linear acceleration = radius * angular acceleration
From the given information, we know that the radius of the circle is 30.2 m. To calculate the linear acceleration, we need the angular acceleration, which we found in part (b). Once we have the value of angular acceleration, we can plug it into the formula to get the linear acceleration.
torque= force*radius
torque*radius= mass*radius^2 * angular acceleration
linear acceleraltion=angularacceleration*radius