The 2-1 b collar C fits loosely on the smooth shaft. If the spring is unstretched when s=0 and t

The 2-1 b collar C fits loosely on the smooth shaft. If the...

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, typically calculated using the expression ½mv², where m represents the mass and v the velocity. Understanding kinetic energy is essential for determining how an object's speed changes when its energy is redistributed within the system, such as when it interacts with a spring.

Conservation of Mechanical Energy

This concept states that in a closed system with no non-conservative forces (like friction), the total mechanical energy remains constant. It involves the transformation between kinetic energy and potential energy. In problems like this, one uses the principle to equate the initial kinetic energy to the sum of the kinetic energy and the stored elastic potential energy when the spring is stretched.

Potential Energy in Springs

Potential energy in springs is associated with their deformation, following Hooke's law which states that the force exerted by a spring is proportional to its displacement from the equilibrium position. The energy stored in a spring can be expressed as ½kx², where k is the spring constant and x is the displacement. Recognizing this concept is crucial for calculating how much energy is absorbed by the spring when the collar moves from the unstretched position.

Transcript

00:01 In this exercise, we have a color that has a massive 2 pounds that is attached to a spring that has a spring constant of 4 pound force per foot.

00:10 And the color is constrained to move vertically.

00:15 And the unstretched length of the spring is equal to 1 foot, as shown here in the figure in blue.

00:24 S is the vertical distance of the color c.

00:31 And we know that when s is equal to zero, that is when the spring is unstretched, the initial speed v0 of the color is equal to 15 feet per second.

00:44 And our goal is to find what is the velocity of the color after it travels a distance of one foot, so after s is equal to one foot.

00:55 So what we need in order to solve this exercise is the concept of elastic force.

01:06 So f, the elastic force, is equal to minus the spring constant k times delta x, where delta x is the distance between the stretched, i'm sorry, is the difference of length between the stretched spring, and the unstructured spring, and the unstraged, stretched spring.

01:36 So it's, i'm going to call it x minus x0, where x0 is the equilibrium position of the spring.

01:43 And this is the information we need in order to solve the exercise.

01:47 So let's apply this to our specific problem.

01:53 So we have that the color is constrained to move in the x direction, meaning that the pole to which the color is attached will exert vertical forces on the the color such that the color will not move in the y direction.

02:17 Okay, so we are, so we can work only with the forces that act in the horizontal direction.

02:29 So consider that this here is the color and the force that the spring exerts on the collar is in this direction shown here in the picture.

02:48 It points along the spring and points downwards.

02:54 That is, it points in the direction of restitution.

02:59 So it tries to bring the collar back to the unstretched position.

03:05 And the horizontal force, horizontal component of this force is pointing in this direction here.

03:19 And it makes a certain angle of theta.

03:22 With the force itself.

03:24 So the horizontal component of the force f x is equal to the elastic force f times the cosine of theta and we saw up here that the cosine of theta is equal to s divided by the square roots of s squared plus one.

04:02 Also the force itself is equal to minus k, that is a spring constant, times the final position of the spring, i'm sorry, the final length of the spring, which is the square root of s squared plus one.

04:25 This is just pythagoras theorem.

04:27 I'm calculating the hypothesis here as a function of the lags minus the unstretched position of the spring, which is one foot...

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