The 10 -lb box falls off the conveyor belt at 5 -ft / s. If the coefficient of kinetic friction

step 1 box that slides down a slope at 5 feet per second. And we want to find the horizontal distance i

The 10 -lb box falls off the conveyor belt at 5 -ft / s. If...

Key Concepts

Kinetic Friction

Kinetic friction is the resistive force acting between surfaces in relative motion. It is quantified by the coefficient of kinetic friction multiplied by the normal force and serves to decelerate moving objects by converting some of their mechanical energy into thermal energy. This concept is central to problems where energy losses due to sliding resistance are considered.

Newton's Second Law

Newton's Second Law, which states that the net force acting on an object equals the mass of the object multiplied by its acceleration (F = ma), is fundamental in relating the frictional force to the deceleration of an object. In scenarios involving friction, this law is used to compute the resulting acceleration (or deceleration) from the given force of friction.

Constant Acceleration Kinematics

Constant acceleration kinematics involves the equations of motion that relate displacement, initial velocity, acceleration, and time. When an object experiences a constant decelerating force such as friction, these equations can be used to determine distances traveled or stopping distances from a given initial velocity and acceleration.

Work-Energy Principle

The work-energy principle is used to relate the work done by forces, including the work done against friction, to the change in kinetic energy of an object. In friction problems, this principle can help determine the distance over which an object decelerates by equating the work done by friction to the loss in kinetic energy.

Transcript

00:01 Box that slides down a slope at 5 feet per second.

00:04 And we want to find the horizontal distance it travels after it leaves point b and right before it hits the car at the very bottom and point c.

00:15 So we're given a few pieces of information.

00:18 So we're given the weight of this box, which is 10 pounds.

00:23 You can convert that to mass by dividing by 32 .2 feet per second squared.

00:29 So for the mass, we get .31 kilograms.

00:35 We'll give, of course, the initial velocity, we'll call v1, and that's 5 feet per second.

00:44 And this little box experiences kinetic friction from point a to point b.

00:50 So the coefficient of that kinetic friction is 0 .2.

00:57 So we know that the horizontal distance x, the equation, will equal the velocity in the x direction, and times time.

01:06 So i want to find the velocity after it leaves point b and the time it takes for that box to reach the bottom at point c.

01:20 So because we're dealing with forces, right, work is force times distance.

01:27 So that box is producing a work and it has a velocity, so it also has kinetic energy.

01:33 So we can use the work energy principle.

01:36 And this part is to find the velocity.

01:43 So we have the initial kinetic energy plus the work being done by the forces equal the final kinetic energy.

01:55 Now let's expand the kinetic energy.

01:59 So we have for t1, we have one half m v1 squared plus for now just leave the work alone is equal to one half m v1, and the final velocity will call bb squared.

02:20 So this is what we're looking for.

02:24 So next let's look at the work being done by the forces.

02:29 So we know that the forces involved are the weight of the box and the kinetic energy or the kinetic friction.

02:39 So first we have the weight of the box.

02:44 This is in the x direction times the distance it travels.

02:51 Minus, well, kinetic friction, the equation is mu k, the coefficient of connect friction, times a normal force times some distance, we'll just call it s .f.

03:07 So to find the weight in the x direction and also to find the normal force, we need to use a free body diagram of the box so that we can see the forces involved.

03:24 So we know we have weight, straight down.

03:30 We can break this into components.

03:32 So you have the y component, wy, and we have the x component pointing to the right, wx.

03:42 And then in the opposite direction of wy, we have the normal force fn.

03:47 And since motion is to the right, the friction is always in the opposite direction.

03:52 So in the opposite direction, you have force due to kinetic friction.

03:59 And to find the components of wx and wy, we are actually giving a little triangle to make the calculations a little simpler.

04:08 So we have this triangle, three, four on the bottom, why for the hypotenuse? so if we're looking at this little triangle, we're looking at weight in the x direction...

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