Two crates, A and B, sit at rest side by side on a frictionless horizontal surface. The crates h

step 1 All right, in this problem, we have two boxes, A and B. They both have their own respective mass

Two crates, A and B, sit at rest side by side on a...

Key Concepts

Free-Body Diagram

A free-body diagram is a simplified illustration that shows all external forces acting on an object. In mechanics, these diagrams help isolate a body and represent forces like gravity, normal force, applied forces, and contact forces, making it easier to apply Newton’s laws. They are essential tools in problems involving multiple bodies, as they help separate and analyze the forces at play for each object individually.

Newton’s Third Law of Motion

Newton’s Third Law states that for every action there is an equal and opposite reaction. This principle explains that forces always occur in pairs acting on different objects. In the context of the problem, while crate A exerts a force on crate B, crate B simultaneously exerts an equal and opposite force on crate A. Recognizing these pairs is crucial when analyzing the interaction between the crates.

Newton’s Second Law of Motion

Newton’s Second Law establishes the relationship between the net force acting on an object, its mass, and its acceleration (F = ma). This law is used to determine the motion of each individual crate by summing forces from the free-body diagrams. It highlights that even if the applied force is small compared to the weight of the crates, the horizontal (net) force determines the acceleration, independent of the unbalanced vertical forces that are counteracted by the normal force.

Frictionless Surface

A frictionless surface is an idealized condition where no frictional force resists the motion of an object along the surface. This simplifies analysis as it allows the problem to focus solely on the applied forces and contact forces between bodies. In this context, since the horizontal surface is frictionless, any net horizontal force directly results in acceleration according to Newton's Second Law, irrespective of the magnitude of the crates’ weights.

Contact Forces

Contact forces occur when objects physically touch each other and include normal forces and any inter-object forces resulting from that contact. In the problem, when crate A pushes against crate B, there is a contact force between them that is governed by Newton’s Third Law. Understanding these forces is important, because even without friction, the interaction between the crates will dictate how they accelerate together as a system.

Transcript

00:01 All right, in this problem, we have two boxes, a and b.

00:04 They both have their own respective masses, massive a, massive b.

00:08 We're pushing on them with this applied force on the left.

00:16 All right.

00:17 When we do that, that is going to generate a contact force from a pushing on b.

00:34 And this contact force here that i've drawn here on a, remember that this is, i could then, technically it's what this force here is what's going to be pushing out.

00:45 Here has a bad f and i apologize so those are our forces um and now we need to draw we're supposed to write ignore this yellow comment for a second we're supposed to write free body diagrams so i kind of already started this um because part of it's really obvious this is the free body a it has weight which is defined by gravity times its mass and then we have the normal response of the ground on the object which is the same thing but positive rather negative because the sum to zero.

01:28 Okay.

01:31 And then we now need to think of left or right forces on a.

01:34 Well, on a, as we just said up there when it was drawing gold, we know pushing to the right, there is the applied force.

01:44 Well, there's also this contact force pushing back against it from b, because as a pushes into b, b is going to push back onto a.

01:51 So we have this force from b, which is the contact force from b on to a.

01:59 And technically, if you want to find that, well, we could take the applied force onto these two objects, divided by the sum of their masses, so as to obtain the acceleration on these pair of objects, because when we're pushing on objects, we're trying to solve this, we can think of them as being one object, because the contact force is an internal force, and we don't really think about that.

02:35 When we're doing a free by diagram, we only do external forces.

02:38 So i could take it.

02:40 The applied force divided by the total mass of both and multiply that because this would give me, sorry, i'm getting ahead of myself, and multiply that times the mass of a, and that would give me the contact force.

02:57 And the reason is, remember that this, when i divide a force of newton's by kilograms, i'm going to get acceleration.

03:06 And then mass of a is a mass.

03:08 So acceleration times mass and that will give me a force in newton's so our forces on our diagram are the the north this one this one this one and this one and what i have here kind of slopely done in red is just some math in case you wanted to know how you would calculate it all right now b is the same thing same thing of its mass times gravity pulling down and its mass times gravity pushing up say some to zero there is no right word force because if you look here, there's no wall over here for it to push against.

03:46 And so this rightward contact force has nothing to retaliate against it or push back.

03:53 And so there is nothing over here.

03:54 We'll leave that blank.

03:55 But there is the contact force from a pushing onto it, this contact force, which is generated from a pushing onto it.

04:09 Make sure i labeled this right.

04:10 Give me one second.

04:11 I apologize.

04:13 Just make sure my labels didn't change on me.

04:18 I think i did that right.

04:40 Okay, so get that.

04:50 Yeah, that looks all good.

04:51 Okay, so that just leaves the one thing we still had, and that was the contact force pushing against it from a, so the contact.