Draw a free-body diagram of a block which slides down a frictionless plane having an inclination

step 1 All right, so on this problem, we have a block on a slope. We don't know what's mass, so we prob

Draw a free-body diagram of a block which slides down a...

Key Concepts

Frictionless Inclined Plane

A frictionless inclined plane is an idealized surface where frictional forces are neglected. This simplification allows the analysis of motion to focus solely on gravitational acceleration and normal forces. By assuming no friction, the net acceleration down the incline is solely determined by the component of gravity acting parallel to the surface, which simplifies calculations and highlights the fundamental principles of motion.

Free-Body Diagram

A free-body diagram is a graphical illustration used to visualize the forces acting on an object. This diagram isolates the object and represents all the external forces, such as gravitational force, normal force, and applied forces, as vectors acting in specific directions. It is a fundamental tool in mechanics to set up equations of motion and analyze the system dynamics.

Gravitational Force Components

In problems involving inclined planes, the gravitational force acting on the object is resolved into two perpendicular components: one parallel to the plane and one perpendicular to it. The parallel component (mg sin?) is responsible for causing the object to accelerate down the incline, while the perpendicular component (mg cos?) is balanced by the normal force, which prevents the object from moving into the surface.

Newton’s Second Law of Motion

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and its acceleration (F = ma). In the context of an inclined plane, this law is applied along the direction of the slope to determine the acceleration resulting from the parallel component of the gravitational force.

Kinematics with Constant Acceleration

Kinematics deals with the motion of objects without considering the causes of motion. When the acceleration is constant, as in the case of a frictionless inclined plane under uniform gravitational force, kinematic equations can be used to relate displacement, initial velocity, acceleration, and time. This allows one to determine the final speed or time of travel for an object starting from rest.

Transcript

00:03 All right, so on this problem, we have a block on a slope.

00:09 We don't know what's mass, so we probably won't be able to figure out the force on it.

00:17 But our first step is we need to draw the free body diagram.

00:23 So there are two forces at play here.

00:28 We have downward force due to gravity, because the box is mass.

00:36 I'm going to label this fg through gravitational force.

00:43 And then there's going to be a force pushing back off on it.

00:47 That's from the ground that holds the box up and make sure it doesn't fall through the floor.

00:57 So that's going to be something at an angle like this.

01:03 This is our normal force.

01:08 You put in.

01:11 And so you can think of, since the normal force is at an angle, you can think of the normal force, which is this way.

01:34 The angle between the normal force and the gravitational force happens to be the same as the angle of inclination here.

01:46 So this right here is theta.

01:54 Because if that angle were zero, then the normal force would be straight up.

02:02 And the normal force would be the same as the gravitational force.

02:07 That's how i think about it.

02:10 And so for the next step, we want to find the acceleration.

02:25 Well, we can't find the force, but we can use geometry to find that the acceleration is relative.

02:34 To the gravitational acceleration.

02:40 The way we do that is we conveniently set our x, y coordinates to be down the ramp is the positive x direction, and perpendicular to the ramp is the y direction.

03:02 And the reason why this is convenient is because we can make the force, or the we can make the acceleration down this to be the acceleration we can make the acceleration down the ramp a component of our gravitational force so we're to draw this out as a triangle let's see uh put g for acceleration since we're solving for acceleration directly so this is g and the acceleration down the ramp is a horizontal component to it.

04:01 So here's the vertical component.

04:13 And then the horizontal component would be something like this.

04:22 So this is a acceleration.

04:26 And then this is, i don't know, g -j hat.

04:35 I'll just call it that...

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