A crate slides down an inclined right-angled trough as in Fig. 5-39. The coefficient of kinetic

A crate slides down an inclined right-angled trough as in...

Key Concepts

Newton's Second Law

This fundamental principle states that the net force acting on an object equals the mass of the object multiplied by its acceleration. It is used to set up equations for motion, particularly when summing forces along specific directions such as along an inclined plane.

Inclined Plane Dynamics

Analyzing motion on an inclined plane involves breaking down the gravitational force into components parallel and perpendicular to the surface. This approach simplifies the study of acceleration and friction, as each component has distinct physical effects on the motion.

Kinetic Friction

Kinetic friction is the force that opposes the movement of an object sliding over a surface. It is directly proportional to the normal force through the coefficient of kinetic friction, which quantifies the interaction between the contacting surfaces.

Gravitational Force Decomposition

On an inclined plane, the gravitational force acting on an object is decomposed into a component along the plane, which drives the motion, and a perpendicular component, which influences the normal force. This decomposition is essential for accurately calculating both the acceleration and the frictional forces involved.

Transcript

00:01 We're going to be looking at problem 30 from chapter 5 of the physics fifth book.

00:06 The question says, a crate slides down an inclined right -angle trough, as in the figure.

00:11 The coefficient of kinetic friction between the crate and the material composing the trough is mu -k.

00:17 Find the acceleration of the crate.

00:19 So this is the kind of setup we have.

00:20 It's a bit difficult to draw, but we've got some trough, which has an angle in there of 90 degrees.

00:29 So that would be 90.

00:30 It's not pretty fair but that'll be 90 and it is perched up at an angle of theta and a crate is sliding down so if we take a look at if we pretend that this whole setup was just down this central line and the crate was resting on that central line we'd have something like this have our angle here theta have our crate there we'd have some weight downwards w and then across have something there so we'd have w plus theta here and w sign theta here i'll write s for the time being w sign theta and then we're obviously going to have our normal force coming out um now what we need to work out is how far how the acceleration of the crate in terms of the various things given so to do that first we're just going to use this diagram here to get a basic equation of what the equation of motion will be so we have ma the rate is which is moving down the force of which it moves down is equal to m g sine theta which is this w sign theta minus the f yeah i forgot about the friction get friction here.

02:08 So subtract the friction.

02:09 And we'll set that as equation one.

02:11 Now we need to take a closer look at what's happening at the trough.

02:13 We have this trough of 90 degrees and we split it down the middle and we are, as you're going to see, have two normal points here.

02:23 We're going to have one going from one side and one going from the other.

02:28 They're both going to have 45 degrees.

02:30 One squared n1, one squared n2.

02:31 And interesting enough, when we have our weight coming down here, we're going to have in this direction mg sign 4 and down here mg cos 45 is obviously if we say for example you 45 is the angle here...

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