A crate of mass M starts from rest at the top of a frictionless ramp inclined at an angle αabove

A crate of mass M starts from rest at the top of a...

Key Concepts

Energy Conservation

This concept is based on the principle that in the absence of non-conservative forces (like friction), the total mechanical energy of a system remains constant. In the problem, the gravitational potential energy lost by the crate as it moves down the ramp is completely converted into kinetic energy, allowing us to relate the height change directly to the final speed at the bottom of the ramp.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is given by the product of the mass, the gravitational acceleration, and the height above a chosen zero potential level. The choice of zero level (reference point) is arbitrary and does not affect the physical behavior of the system, as only differences in potential energy are physically meaningful.

Choice of Potential Energy Reference Level

The reference level for potential energy can be chosen arbitrarily without affecting the outcome of the energy conservation equation. Whether taking the zero potential energy level at the bottom or the top of the ramp, the difference in potential energy over the height change is the same, leading to an equivalent expression for the speed at the bottom. This illustrates the concept that physics does not depend on the coordinate system or reference point chosen.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion, given by one-half the mass times the square of the velocity. In the context of this problem, as the crate descends the ramp, its increase in kinetic energy comes from the conversion of gravitational potential energy into kinetic form, and this relationship is used to solve for the speed at the bottom.

Normal Force

The normal force is the force exerted by a surface perpendicular to it, preventing objects from penetrating the surface. In work or energy calculations, the normal force does no work on the crate because its direction is perpendicular to the displacement of the crate along the ramp. Since work is the dot product of force and displacement, only the components of force parallel to the displacement contribute to the work done on the object.

Transcript

00:01 Problem 7 .6, we have a crate of mass m that starts at the rest at the top of a frictionous ramp, inclined in angle a or alpha above the horizontal.

00:11 Find its speed at the bottom of ramp the distance deep from where it started.

00:14 And it asks us to do us in two ways.

00:17 So the first way, you're just going to set the potential energy to be zero at the bottom of ramp.

00:21 So in order to do that, i'm going to write y equal zero here.

00:27 And then therefore we will be measuring the potential energy with respect to this point.

00:34 So if the object is at this bottom of the ramp here, it's kind of potential energy of zero.

00:39 So in order to do that, we're going to find speed of the bottom ramp using work.

00:46 So we're going to write work due to gravity, and that's going to be equal to the change in the height of the object.

00:56 So we're going to write m, g, and y2 minus y1.

01:06 Okay, so here we have y1.

01:09 That's going to be where it ends.

01:11 So we know that's going to be zero, but we need to find an expression for y2.

01:16 So y2 is just going to be the position of the block at the top of the ramp.

01:20 That's going to be its vertical displacement from the bottom with respect to the y equals 0 at the bottom.

01:26 So we need to find an expression and what we have available is alpha and d.

01:32 So what i can do is i can write that y2 is equal to d sine of alpha.

01:47 So just from your basic knowledge of triangles, in order to calculate what that distance here is from the top of, from the top of that triangle here to the right angle here, you're going to take the hypotenuse and multiply by sign of this angle here.

02:05 So that's going to be d -sign theta.

02:07 So in the end here we're going to get m -g -d -d -sign alpha.

02:22 Okay.

02:24 So that's going to be then equal to the negative of the difference of u of gravity.

02:34 And what we can do now is we're going to use that in our conservation of energy.

02:40 Equations so we have k1 plus u1 equal to k2 plus u2 sorry about that so we have that it starts at rest so that means that this is zero this k1 is zero we know what the change in the potential energy is so that's going to be equal to u1 minus u2 so we can just rearrange this equation to be u1 minus u2 equal to k2.

03:23 Then we get mgd sine alpha, mgd sine alpha, equal to k2.

03:38 And then from there, we know what the expression is for kinetic energy of a single object.

03:45 So then that's just going to be equal to one half m v squared...

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