Neglecting the mass of a spring

[Calpalned]

Homework Statement

Homework Equations

See above

The Attempt at a Solution

This is the textbook's explanation...

1) Why does it say $k_{speed} = K_{mass} +$... and not $k_{total} =$...?

2) If force isn't constant, then work requires an integral, which in turn means that kinetic energy would require one too, right? An example is the KE for the spring.

3) If (2) is true, then why do we take the integral for the kinetic energy of the spring, but not for the mass? The only force on the mass in the horizontal direction is that of the spring, which isn't constant.

4) In the kinetic energy equation, why does the mass get $v_0$ but the spring gets $v$? Doesn't the mass move at the same rate as the spring? The question statement indicates that "each point on the sring moves with a veolocity proportional to the distance..." Thus, when the spring is fully stretched, the mass moves at a veloctity $v_0$. When the spring has half contracted, the speed reduces to $\frac{v_0}{2}$

5) For the spring, we take the integral of $v$ with respect to $dm$ (presumably because that's what the hint says). I don't understand why it's $dm$ - mass doesn't change. Could we also get the same answer with $m$, $dv$?

Thank you so much!

 

[BvU]

1) No particular reason. They mean the same thing as you.
2) "which in turn means" is not justified. Work is an integral. In simple cases that integral can be a simple product.
Kinetic energy is an integral if not all constituents move with the same speed.
3) see 2.
4) The spring doesn't get v. The little piece of the spring dm from x to x+dx from the fixed end gets speed v. And that v depends on x, hence the integral.
When the spring is fully stretched, nothing moves.
At all times during the oscillations, the left end of the spring has speed 0. The right end moves at speed v₀ (that is NOT a constant! -- see the problem 68 problem statement ). And the halfway point at $v_0\over 2$.
5) No. We integrate the contributions $dK_{\rm spring} = {\tfrac 1 2} v(x)^2\; dm$ of the small sections of the spring that all move at different speeds and have mass $dm = {M_{\rm s}\over D}\; dx$ when the spring length is D (also: not a constant in time !)

 

[haruspex]

Why does it say k_speed

It doesn't, it says k_{speed v0}. I.e. the KE of the system when the mass is moving at speed v₀.

The only force on the mass in the horizontal direction is that of the spring, which isn't constant.

The equation is for an instant in time. At any instant, the force is constant along the spring.

Note that where the problem statement says "Each point of the spring moves with a velocity proportional ..", that is part of the specification of the problem. It is not a general fact about oscillations of a spring with mass. More generally, there may be vibrations within the spring.