A 500-g collar can slide without friction along the...

Name: Problem 65, Chapter 13: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-07-20T20:53:34-08:00
Duration: 5 min 19 s
Channel: Eric Mockensturm
Description: A 500-g collar can slide without friction along the semicircular rod BCD. The spring is of constant 320 N/m and its undeformed length is 200 mm. Knowing that the collar is released from rest at B, determine (a) the speed of the collar as it passes through C, (b) the force exerted by the rod on the collar at C.

A 500-g collar can slide without friction along the semicircular rod BCD. The spring is of constant 320 N/m and its undeformed length is 200 mm. Knowing that the collar is released from rest at B, determine (a) the speed of the collar as it passes through C, (b) the force exerted by the rod on the collar at C.

Key Concepts

Force Analysis on Constrained Motion

In scenarios where an object is confined to move along a predetermined path, such as a frictionless rod or a track, analyzing the forces acting on the object involves decomposing them into components normal and tangential to the path. The normal force provided by the constraint (in this case, the rod) ensures that the path is followed, and its magnitude often relates to the need for centripetal acceleration.

Circular Motion and Centripetal Force

When an object moves along a curved path, such as a circle, it experiences an inward (centripetal) force that keeps it on the path. This force is related to the object's mass, its speed, and the radius of the curvature of its path. Understanding centripetal force is crucial for analyzing the dynamics of objects in constrained circular motion.

Hooke's Law and Spring Potential Energy

Hooke's Law relates the force exerted by a spring to its displacement from equilibrium, with the spring force being proportional to the extension or compression. The corresponding potential energy stored in the spring is calculated from the spring constant and the amount of deformation. This concept is fundamental when analyzing systems where elastic forces affect the motion of a mass.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy of a system remains constant. In problems involving a mass-spring system or objects moving along a fixed path, the conversion between kinetic energy and potential energy (both elastic and gravitational, if applicable) is used to determine quantities like speed at various points in the motion.

Transcript

00:01 Now we have a collar writing on a circular rod or semicircular rod.

00:09 And there's a spring attached to the collar and also to a point attached to ground.

00:17 And the semicircular rod is now in the vertical plane.

00:23 So we have to be careful and we have to make sure we account for changes in gravitational potential.

00:29 So this is kind of in a three -dimensional.

00:32 Mentioned so it's easiest to write position vectors and i use point c as the origin as syndicated in the book and so then point a we have and point b we have and then point b you have the position to all of those points the position vector we're told the mass of the slider is a half kilogram and the spring force the spring constant is 320 meters per meter and the spring has an unstretched length of 200 millimeters.

01:08 And so we're told that at position b, the slider is, the collar is released from rest, so that the kinetic energy is zero there.

01:19 The potential energy at b has potential energy from the spring and from gravity.

01:26 So yb is just the y component of r, so b.

01:31 And with a length from a to b, we can figure that out because we know the position of a and with the position of b, so we can subtract those two and find the magnitude would be the length.

01:47 And so we have this.

01:49 And then at c, again, we have the potential energy.

01:55 There's no gravitational potential at c because y of c is zero.

02:00 And we can figure out the length of the distance from a to c, so the length of the spring, because we know point a and point c, and then we have some kinetic energy at that point.

02:11 And then at point d, we have, again, the length of the spring we can figure out because we have a and b.

02:20 And we have some gravitational potential again.

02:23 And at point d, we have y, sub d is 150 millimeters.

02:29 And then we also have some kinetic energy there...

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