The spring has a stiffness k=200 N / m and an unstretched length of 0.5 m. If it is attached to

The spring has a stiffness k=200 N / m and an unstretched...

The spring has a stiffness $k=200 \mathrm{N} / \mathrm{m}$ and an unstretched length of $0.5 \mathrm{m}$. If it is attached to the 3 -kg smooth collar and the collar is released from rest at $A$ determine the speed of the collar when it reaches $B$. Neglect the size of the collar.

Key Concepts

Hooke's Law

Hooke's Law describes the linear relationship between the force exerted by a spring and its displacement from equilibrium. It states that the force is equal to the stiffness (spring constant) multiplied by the displacement, which is fundamental when analyzing the behavior of springs in mechanical systems.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is either stretched or compressed. It is calculated using the formula ½ k x², where k is the spring constant and x is the displacement from the unstretched length, and it plays a key role in converting stored energy into kinetic energy during motion.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In systems involving springs and moving masses, the potential energy stored in the spring is converted into the kinetic energy of the mass, making energy conservation a crucial concept for solving dynamics problems.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is calculated as ½ m v², where m is the mass of the object and v is its velocity. Understanding kinetic energy is essential for determining the speed of a mass after energy transformation has occurred.

Transcript

00:01 We have a collar attached to a rod, which is also attached to a spring.

00:05 We want to find the velocity, which we call vb, when the collar reaches the bottom, at point b.

00:16 So we're told that k, the stiffness of this spring, is 200 newton meters, and the length of this spring, when it's unstretched.

00:30 We'll call that just s1 .5 and of course the mass of collar just three kilograms since you're dealing with the spring and is being stretched or compressed then there has to be some sort of spring potential and this collar will move at a certain speed which is what we're looking for so there's a kinetic energy involved so we can use the conservation of energy equation so its initial kinetic energy is zero because the caller starts at rest.

01:04 So we just have the potential initial, which is mg times h .a, so the height when it's at point a, plus the potential of the spring when it's at point a, so one half k, x, a squared.

01:22 So the distance of the spring when it's at point a equals the final, velocity when it reaches point b so 1 half m vb squared plus the potential only of the spring because when it reaches the bottom the height is zero so the spring potential here is one half k times xb squared so this is the distance of the spring when it reaches point b so to find xa and xb we can draw a triangle from the diagram so we know that the height when it's at point b is two meters and we know that the x is 1 .5 meters so we can find a hypotenuse you can call that c and the z and the hypotenuse here is actually the length of the spring at point b or point a so z is just two squared plus 1 .5 squared square root and and z is 2 .5 meters.

02:46 So that means xa, the distance of the spring at point a is that distance 2 .5 and z minus the length of the spring when it's uncompressed, which is given as 0 .5.

03:05 So that means for xa, we have 2 meters.

03:13 So that's the length of the spring at point a...

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