A 2100-kg airplane pulls two gliders, the first of mass 340 kg and the second of mass 280 kg,

A 2100-kg airplane pulls two gliders, the first of mass 340...

A 2100-kg airplane pulls two gliders, the first of mass $340 \mathrm{~kg}$ and the second of mass $280 \mathrm{~kg}$, down the runway with acceleration $1.7 \mathrm{~m} / \mathrm{s}^{2}$ (Fig. 4.22). Neglecting the mass of the two ropes and any frictional forces, determine the magnitudes of (a) the horizontal thrust of the plane's propeller, (b) the tension force in the first rope, (c) the tension force in the second rope, and (d) the net force on the first glider.

Key Concepts

Free-Body Diagrams

Free-body diagrams are graphical representations that depict all the forces acting on a single object. They are instrumental in breaking down complex problems into simpler parts, allowing one to identify and isolate forces such as thrust, tension, and net force, and then apply Newton's second law to analyze the motion of each component in the system.

Newton's Second Law

Newton’s second law is the cornerstone for analyzing the motion of objects and states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is used to determine the necessary forces, such as thrust or tension, that result in a specific acceleration of a mass or system of connected masses.

Tension in Ropes

Tension refers to the pulling force transmitted along a rope, string, or cable when forces are applied at its ends. In problems like this, where multiple bodies are connected by ropes and subject to a net external force, the tension force acting in each rope is a crucial intermediate force that connects the dynamics of each object in the system.

Systems of Connected Bodies

When multiple bodies are linked together by ropes, they form a connected system where the motion of each body is subject to shared acceleration. Analyzing the system as a whole can simplify the determination of overall forces, while examining individual bodies using free-body diagrams allows for the calculation of specific forces like tension within the connecting ropes.

Transcript

00:01 Okay, we're looking at problem 4 .55, where we have two gliders being towed by an airplane.

00:12 And we want to know about the force from the propeller and then the tensions.

00:20 So i'm not going to attempt to draw these.

00:23 So i'll just draw them as circles.

00:31 Okay, pretend those are airplanes.

00:32 So we've got these ropes.

00:37 In between and then let's call these let's say they have mass m2 m1 and mp for plane so the first one is the plane and then we have the first and the second glider and this is kind of the setup you got this plane goes that way pull the gliders with it okay so we first want to look at oh, and they give us the masses, and we know that it accelerates at 1 .7 meters per second squared.

01:17 Okay.

01:19 So we want to know the horizontal thrust of the plane's propeller, and then the tensions.

01:25 So i'm going to kind of do this all in one.

01:30 I'm kind of going to do it all at once, just because i think that's the easiest way to do it.

01:36 So what we want to do is just look at the forces on each aircraft by itself.

01:46 So here's our m2, here's our m1, and here's our plane.

01:51 So on, let's do the plane first.

01:54 On the plane, we have the force from the propeller that pushes it forward.

02:01 And then we have the tension from the rope that's connected to the glider, and that's pulling it back.

02:08 Let's call that tension t1.

02:15 And then we have like gravity and the normal force while they're on the ground.

02:22 In the air we'd have like lift and gravity.

02:25 But we're just going to look at the x forces, the forces in the horizontal direction.

02:31 So on the first glider, we have the tension from the rope.

02:39 That's pulling the glider.

02:40 And we have the tension from this other rope that's pulling back on the glider.

02:49 On this last glider, we only have the tension of that second rope.

02:55 Okay, so let's start with the second glider.

03:01 So for m2, i'll just write all these down here, m1 and mp.

03:10 So sum of the forces equals the mass times the acceleration.

03:17 The mass here is the mass of m2, since we're just looking at the forces on m2.

03:25 So we have t2 equals m2a.

03:32 Okay, for m1 we have the sum of the forces equals m1a.

03:43 T1 minus t2 equals m1a.

03:51 Same for the plane, sum of the forces.

03:54 It's m -p -a.

03:58 So the propeller force minus the tension equals the mass to the acceleration.

04:06 Okay.

04:08 Now what we want to do is, since we're solving for in part a, we're solving for f.

04:17 That's the force of the propeller.

04:19 So force of the propeller depends on tension, t -1, and t -1 depends on t -2.

04:26 So we're going to have to do some plugging in here.

04:30 So if we take our tension one equation and say tension one, so we have tension one minus tension two, looking at this equation here, equals m1a.

04:47 So we're solving for t1...

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