Block A has a weight of 60 lb and block B has a weight of 10 lb. Determine the speed of block A

Block A has a weight of 60 lb and block B has a weight of 10...

Key Concepts

Conservation of Energy

This concept states that in an isolated system, the total energy remains constant. In the context of mechanical systems, it is the balance between gravitational potential energy and kinetic energy. Understanding this principle allows one to determine how a loss in potential energy due to a change in height results in a gain in kinetic energy, which can then be used to calculate speeds or velocities.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. In energy transformation problems, the conversion of potential energy into kinetic energy is key to finding the speeds of objects after they have moved. This concept is essential when applying the work-energy theorem to mechanical systems.

Gravitational Potential Energy

This is the energy stored in an object due to its position in a gravitational field. Changes in height relative to a reference level result in changes in gravitational potential energy. In the analysis of motion on inclines or vertical movements, determining these changes is crucial as they are directly converted into kinetic energy in the absence of friction.

Constraint Relationships in Pulley Systems

In systems involving pulleys and inextensible cords, the movements of the connected objects are interdependent. These constraints govern how a displacement in one part of the system results in a corresponding displacement in another, affecting the overall energy distribution and motion analysis.

Transcript

00:01 In this problem, we have two blocks, a and b, and we want to find the velocity of block a, which we'll call b a.

00:11 And it moves down to the plane at a distance of 5 feet, and we'll call that s .a.

00:21 And of course, we're given the weight of block a, and that's 60 pounds.

00:29 And you can convert that to mass, divided by 32 .2 feet per cent.

00:35 Second so it was the second squared so our mass a is 1 .86 kilograms the weight with block b is 10 pounds and we can convert that again so we get m b is equal to 0 .31 kilograms so since we're dealing with the distance of 5 feet and we're dealing with forces on the blocks and we're also we also have speed involved, we can use the work energy principle to find the velocity va.

01:20 So the way this works is we're given the equation kinetic energy initial of the event plus the work being done is equal to the final kinetic energy.

01:34 And since the blocks are at rest when it begins, then the initial kinetic energy is zero.

01:42 And so we're left with the work being done is equal to the final kinetic energy.

01:54 We can simplify that.

01:59 So work being done is equal to 1 1 .5 .m.

02:07 V .a.

02:09 Squared plus 1 .5 .m .b.

02:14 B squared.

02:16 So now let's look at the equation for work.

02:21 So we want to look at the work being done by both block a and block b.

02:32 So we're given w .a, the force of block a times the distance s, in this case sa, minus weight of block b, the force and block b times its distance that it travels sb.

03:02 Block a is at an angle.

03:04 And so we can't just say weight a or the force time the distance because it's at an angle so you want to find the force at that angle and so we're given a triangle like this five four at the bottom three on the side so we want to find the force in this x direction where this four is at so we need to incorporate this three and five so it's this equation becomes weight with block a times the ratio of three -fifths times s a minus wb times sbb okay now we plug that back in to our equation this one so now we have w a three -fifths times s .a.

04:22 Minus wb.

04:25 S .b.

04:26 S .b.

04:27 Is equal to one -half.

04:28 It stays the same.

04:29 M .a.

04:31 B -a -squared.

04:34 Plus one -half.

04:36 M -b.

04:39 B.

04:40 So looking at this equation we have, we know weight a...