A model airplane of mass 0.75 kg is flying at constant speed in a horizontal circle at one end

A model airplane of mass 0.75 kg is flying at constant...

A model airplane of mass $0.75 \mathrm{~kg}$ is flying at constant speed in a horizontal circle at one end of a 33 -m cord and at a height of $18 \mathrm{~m}$. The other end of the cord is tethered to the ground. The airplane makes $4.4$ rev/min and the lift is perpendicular to the unbanked wings. (a) What is the acceleration of the plane? (b) What is the tension in the cord? ( $c$ ) What is the lift produced by the plane's wings?

Key Concepts

Uniform Circular Motion

Uniform circular motion is the movement of an object along a circular path at constant speed, where the direction of velocity continuously changes. This change in direction produces an acceleration known as centripetal acceleration, which is directed toward the center of the circle. The concept involves understanding how angular velocity, period, and radius determine the magnitude of this centripetal acceleration and the net force acting on the object.

Centripetal Acceleration and Force

Centripetal acceleration is the acceleration required to keep an object moving in a circular path and is given by the relation a = ?²r or a = v²/r, where ? is the angular velocity, v is the tangential speed, and r is the radius of the circle. The corresponding centripetal force is obtained by multiplying this acceleration by the mass of the object, and it is this inward force that maintains the circular trajectory.

Force Decomposition

In many physics problems, forces acting at an angle are resolved into perpendicular components (typically horizontal and vertical) to simplify analysis. For an object in circular motion under an external constraint, such as being tethered, the tension force in the tether can be decomposed into a vertical component, which often counteracts gravitational forces, and a horizontal component, which provides the necessary centripetal force for the circular path.

Aerodynamic Lift

Aerodynamic lift is the force generated by an object, like an airplane wing, moving through air at an angle relative to the airflow. It typically acts perpendicular to the direction of motion and is essential for sustaining flight by counterbalancing gravity. In dynamic problems, lift must sometimes be balanced with other forces to satisfy equilibrium conditions in different directions.

Transcript

00:01 So to start this question, we'll look at drawing what's happening here in the diagram.

00:07 So from a top -down view, we've got this dotted line, which is the circle that the plane travels in, and this circle has some radius r, which we calculate.

00:20 And it has some, just in this direction, it has some mv squared over r to the right, which is a centripetre acceleration.

00:28 The diagram showing it side on is more interesting to see what happens because we got to see the forces.

00:38 So we've got mg.

00:40 So this is our plane here and it's going to be going around like this, something like that.

00:50 And so we have some weight mg going downwards, our centrifugal acceleration going inwards to the circle.

00:59 This is our green line is our string with some tension going downwards and we have our lift force fl going upwards.

01:08 Now what do we know about this system? well, we know that the cord is at a height of 18 meters and it has a length of 33 meters.

01:17 So doing that we can work out what r is which is the radius of the circle and we can also work out what theta is.

01:24 For starters, for part a, it asks us to work at the acceleration plane.

01:30 So first of all, let's get our omega, because obviously we're going to be doing v is equal to r.

01:36 Sorry, v is equal to omega r.

01:41 So we have 4 .4 revolutions a minute.

01:44 Divide that by 60, we have 0 .0 .073 revolutions per second, and that's our f...

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