In Fig. 6-59, a crate slides down an inclined right-angled trough. The coefficient of kinetic fr

step 1 Okay, so here we've got this block sliding down this trough. And we can kind of treat this as a

In Fig. 6-59, a crate slides down an inclined right-angled...

Key Concepts

Inclined Plane Dynamics

This concept involves the analysis of objects moving on a sloped surface where gravitational force is decomposed into components parallel and perpendicular to the incline. It serves as a classic model in mechanics that allows the study of motion along a fixed angle and the effect of forces on an object moving along a curve or track.

Force Decomposition

In analyzing motion on an incline, the gravitational force acting on an object is split into two components: one parallel to the surface, which is responsible for the motion down the plane, and one perpendicular, which determines the normal force. This decomposition is critical for solving problems involving friction and acceleration.

Kinetic Friction

Kinetic friction is the resistive force that acts opposite to the direction of motion when an object slides against a surface. It is calculated using the coefficient of kinetic friction multiplied by the normal force. Understanding kinetic friction is essential for determining the net acceleration when frictional forces are significant.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and acceleration. It is used to derive expressions for acceleration by summing the forces along the direction of motion, including both gravitational components and frictional forces. This law is fundamental in translating the physical forces into mathematical expressions for acceleration.

Transcript

00:01 Okay, so here we've got this block sliding down this trough.

00:05 And we can kind of treat this as a normal sliding problem to start where we have the downward weight force that is acting straight down, and this is going to be split into parallel and perpendicular components.

00:19 We're going to have a friction force that is up the ramp, and then we're going to have a normal force that is perpendicular to the ramp.

00:30 So if we write the forces acting on this thing, we're going to have the mass of it times its acceleration equals the parallel component of the weight force, so mg sine a theta, minus that friction force.

00:48 And what is this friction force? well, the friction force is going to be equal to mu kinetic friction times the normal force.

00:59 However, this is complicated because we're actually going to have two different normal forces acting on this.

01:06 We're going to have a combined normal force, and we're going to have a friction force due to each side of the trough.

01:23 And this coefficient of kinetic friction is given to us in terms of the trough itself.

01:28 So we are going to have this equation here where we have twice the friction force.

01:34 We're going to have a little bit of friction over here.

01:36 We're going to have some friction over here.

01:38 So now we've got our friction forces, 2 mu kinetic friction times the normal force acting on each side.

01:48 Well, now we need to figure out what that is.

01:52 For that, we need to look at what is the combined normal force here? well, the combined normal force, and let's just call that fnc for combined, that's going to be equal to the vertical part of each of these because their horizontal parts are going to cancel each other out.

02:15 And so we're going to get two times their components cosine of theta, where this theta, that's not actually theta.

02:32 That's 45 degrees.

02:36 So cosine of 45 degrees.

02:38 Well, 2 times cosine of 45 degrees, that's going to give us square root of 2.

02:45 So we'll just call this square root of 2 times this normal force there.

02:50 Well, this is the same as this.

02:52 So i can then plug and solve that for that.

02:57 We're going to get this over square to 2, or square to 2, fn, c, over.

03:04 2.

03:05 Plug this into this.

03:06 The 2s are going to cancel...