A 2.50-kg block is placed on a rough surface inclined at 30^∘ . The block is propelled and launc

A 2.50-kg block is placed on a rough surface inclined at...

Key Concepts

Free-Body Diagram

A Free-Body Diagram is a schematic representation of all the forces acting on an object, isolating the object and representing forces such as gravity, normal force, friction, and any applied forces. In problems involving inclined planes, it’s essential to illustrate how these forces decompose along the plane and perpendicular to it, which helps in setting up the relevant equations of motion.

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 its acceleration (F_net = m*a). In the context of an inclined plane, this law is applied by resolving forces along chosen axes (one parallel and one perpendicular to the incline) and setting the sum of the forces along each axis equal to the mass times the corresponding acceleration (or zero in the direction of no acceleration).

Gravitational Force Decomposition

Gravitational force decomposition involves breaking down the weight of an object (mg) into components that are parallel and perpendicular to an inclined surface. The parallel component (mg sin ?) influences motion down the plane, while the perpendicular component (mg cos ?) is balanced by the normal force. This decomposition is crucial for setting up the equations of motion on an incline.

Kinetic Friction

Kinetic friction is the resistive force that opposes the motion of an object sliding across a surface and is proportional to the normal force acting on the object. It is quantified as f_k = ?_k * N, where ?_k is the coefficient of kinetic friction. Recognizing the role of kinetic friction is essential for analyzing motion when non-conservative forces are present.

Kinematic Equations

Kinematic equations relate the parameters of motion—including displacement, velocity, acceleration, and time—under constant acceleration. These equations, such as v^2 = v_0^2 + 2ad, are used to determine unknown quantities like acceleration when an object’s initial velocity, final velocity, and displacement are known, and are particularly useful in problems where an object comes to rest after a given distance.

Transcript

00:01 Here, somebody has propelled a block down an incline with a speed of 1 .6 meters per second.

00:10 And the block slides a distance of 1 .10 meters before coming to a stop.

00:17 And the question is, what is the coefficient of kinetic friction on that surface? so to answer that, we are going to need both kinematics and newton's second law.

00:35 So a reminder of what those are, and then we'll get into the kinematics, which usually is a little bit easier to do.

00:43 But newton's second law is the sum of the forces equals mass times acceleration.

00:51 And there are three equations usually that you use to work with constant acceleration.

00:59 I am going to concentrate on the one equation that does not involve time.

01:06 Because i see that no time is given in the problem.

01:11 So i'm going to use the v final squared minus v initial squared is equal to twice the acceleration times the distance.

01:25 And we're going to pick a coordinate system with y pointing up and x pointing down along the incline.

01:41 And with that as our situation, we have v.

01:48 Final squared zero, and of course the initial is 1 .60 meters per second, which we will square, is 2 times the unknown acceleration, times the distance of sliding 1 .10 meters is down the incline.

02:10 So what this will provide for us is that the acceleration is negative, and specifically it is in the minus x direction.

02:21 And kind of working that out, it winds up to be 1 .60 squared with a negative in front, divided by twice 1 .10.

02:35 And let me sublabel that with an x.

02:42 So i realize that that is an acceleration along the incline.

02:52 Henceforth stubbed the x direction.

02:56 And that comes out to be minus 1 .16 meters per second squared.

03:05 And why i did that first is knowing that newton's second law is going to involve the acceleration.

03:12 And typically what you want to do is break this into two of equations, one for the x direction and one for the y direction.

03:23 So decompose your vectors.

03:27 And you really can't use newton's second law until you have a force diagram, and then you can start working with the components.

03:38 So let's create a force diagram.

03:48 So i have my mass, which i know, and i'm going to create a weight downward.

03:55 That's always a good place to start.

04:00 And i'm going to show that that weight acts at an angle.

04:07 So i'm going to kind of show my axes in there.

04:10 I have an x -axis and a y -axis.

04:16 Let's make that a different color.

04:20 And my weight actually makes an angle of 30 degrees to my y -axis.

04:28 That is, it becomes zero.

04:30 Zero if i am flat on a horizontal, completely horizontal, oriented surface.

04:41 So that's the easy one.

04:43 The other easy one to draw is the normal force is straight along the y -axis.

04:56 And finally, there is a force of friction, which is going to act opposite the motion, and it is sliding friction.

05:05 So we will call that f sub k.

05:10 And now i can decompose things into components and set up my newton's second law.

05:20 So i usually make a column for my sum of fx and another column for my sum of fy...

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