The 10-16 collar starts from rest at A and is lifted by applying a constant vertical force of F=

The 10-16 collar starts from rest at A and is lifted by...

Key Concepts

Rotational Kinematics

Rotational kinematics involves the study of objects moving along curved paths, where angular variables such as angular displacement, angular velocity, and angular acceleration are used. By relating these angular quantities to the corresponding linear measures, one can analyze the motion of objects undergoing rotation or moving along curved trajectories.

Force Resolution and Geometric Analysis

In dynamics, especially for systems experiencing both translational and rotational motion, it is important to resolve forces into components aligned with the direction of motion. The geometric relationship between the applied force and the path of the moving object determines the effective component of the force that does work, which is pivotal for correctly calculating power in such scenarios.

Work and Power

Instantaneous power is the rate at which work is done by a force. It is calculated as the dot product of the force vector and the velocity vector at the point where the force is applied. This concept is key when determining how quickly energy is being transferred to or from a system as it moves.

Transcript

00:01 So in this problem, we have a collar that weighs 10 pounds and is being moved up the rod by a constant force of 25 pounds.

00:12 And we want to know that the power that this generates as soon as the angle is 60 degrees.

00:21 So we can also convert the weight to mass by dividing by 32 .2 feet per second squared.

00:28 So for the mass of the collar, we have .31 kilograms.

00:34 Since we're dealing with power, we know that a variation of the equation of power is the force times the velocity.

00:44 But in this problem, we have the force at some angle and the velocity is going up straight up the rod.

00:56 And the angle it creates the 60 degrees.

01:00 So power becomes force and velocity times cosine of that angle theta.

01:11 So we know the force and we know the angle.

01:15 So what's missing is the velocity.

01:19 So to find the velocity, we can use the work energy theorem to find the velocity.

01:28 Since we're dealing with movement, that means that there's a kinetic energy involved.

01:32 And because the movement is due to a force, that means there's work going on as well.

01:39 So that's what we can use the work energy theorem, which is the initial kinetic energy plus the work done by the forces is equal to the final kinetic energy.

01:55 But in this problem, the caller starts at rest.

01:58 So the initial kinetic energy is zero because of that instant velocity is zero.

02:06 So then this becomes, or so just leave work alone for a second.

02:12 So work is equal to, and then final kinetic energy is one half m v2 squared.

02:20 And this is the velocity we're trying to solve.

02:23 We can call this equation one.

02:28 So again, from this problem, we know the mass, but we don't know the work involved.

02:34 So now let's find the work.

02:37 And we know that work is force, ten times.

02:41 Distance and the forces involved here is the is the force of 25 pounds and also the weight of a collar so we have the force 25 pounds times the distance it travels but that's what we don't know as well so if we think about it initially we have a triangle right we have the height four and we have the distance x in the x direction three so we can find the hypotenuse, 4 squared plus 3 squared, square root, and we get 5.

03:31 So this is the position initially, but then it moves until it reaches 60 degree, an angle of 60 degrees.

03:42 So we don't know the vertical.

03:45 It's called that y2.

03:48 The x direction still remains the same three, and the hypotenuse changes as well.

03:55 We can call that l2...

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