Calculate Kinetic Energy of Spring with Mass M & Speed v

In summary, the conversation discusses the calculation of the kinetic energy of a spring with mass M, equilibrium length L0, and spring constant k. This involves dividing the spring into smaller pieces and finding the speed and mass of each piece, and then integrating to find the total kinetic energy. The relationship between k and kinetic energy is given by the equation k=1/6Mv^2. The question also asks about the speed, mass, and kinetic energy of a middle slice of the spring at a specific location x. To find the total kinetic energy, all the individual kinetic energies of the slices must be added together through integration.
  • #1
adrian116
43
0
the question is following:
To find the effect of the spring's mass, consider a spring with mass M, equilibrium length L0, and spring constant k. When stretched or compressed to a length L, the potential energy is (1/2)Kx^2, where x=L-L0.
a) Consider a spring, as described above, that has one end fixed and the other end moving with speed v. Assume that that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v. ( Hint: divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not
(1/2)Mv^2, since not all of the spring moves with the same speed.)

my director instructed us that : k=∫dk=1/2ρdl(l/L*v)2=1/6Mv2
i don't understand how to work out this 1/2ρdl(l/L*v)2
also, what is the relationship of k with k.e.?
 
Last edited:
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  • #2
If the free end of the spring is moving at speed v_L,
and the fixed end is moving at v_0 = 0 , then
how fast is the middle slice (at x=L/2) moving? _____
What is the middle slice's v^2 ? ______
If the middle slice is dx long, what is its mass? ______
what is the middle slice's Kinetic Energy? ______

Now look at a slice at more general location x between 0 and L .
add the Kinetic Energy of all these general slices by integrating.
 

What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion.

How is kinetic energy calculated?

Kinetic energy is calculated using the equation: KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

What is the unit of kinetic energy?

The unit of kinetic energy is joules (J) in the SI system of units.

Does the mass or speed have a greater impact on kinetic energy?

The speed of an object has a greater impact on its kinetic energy compared to its mass. This is because kinetic energy is directly proportional to the square of the object's velocity.

Can the kinetic energy of an object be negative?

No, the kinetic energy of an object can never be negative. It is always a positive value as it represents the energy of an object in motion.

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