/sci/ - Science & Math

[Anonymous No.4259687]

/sci/ confirm for me:

In a pendulum, tension force is maximum when the pendulum is in equilibrium position, and it's equal to ma+mg.

Also, can you tell me at what position would tension force be minimum in a pendulum, and what that value would be?

Mucho gracias.

[Anonymous No.4259692]

bump

[Anonymous No.4259698]

nobaday?

[Anonymous No.4259704]

The point where momentum = 0, I imagine.

[Aquila !!gAQL6Rry0wi No.4259727]

Equilibrium, as in not moving (<span class="math">F_{net}=0[/spoiler])? Or as the pendulum is moving and <span class="math">\theta=0[/spoiler]?

[Anonymous No.4259730]

i think it is not ma + mg

[Anonymous No.4259739]

The tension is equal to the force along the direction of the pendulum.
Since the pendulum bob is bound to the string in a circular path it must have a centripetal acceleration which is acted upon the bob by the string, and hence by Newton's third law the bob pulls on the string with the same force (opposite).

So the tension from gravity at any angle is simply:
<span class="math">mg cos(\theta)[/spoiler]

The tension from the centripetal acceleration is (simply from definition):
<span class="math">m\omega^{2}r[/spoiler]

So the tension is:
<span class="math">mg cos(\theta)+m\omega^{2}r[/spoiler]

The first term is a maximum when <span class="math">\theta=0[/spoiler] and the second term is a maximum when <span class="math">\omega[/spoiler] is a maximum. Both of these conditions occur at the equilibrium point just like you said.

Now, the first term continually decreases with <span class="math">\theta[/spoiler] and won't start increasing again until <span class="math">\theta=\pi[/spoiler] which would be if the pendulum went ALL the way around, but this doesn't happen in a pendulum, so the minimum is at the ends of the pendulum.

Same for the second term, it is a minimum when <span class="math">\omega[/spoiler] is a minimum. Considering that <span class="math">\omega[/spoiler] changes direction, this obviously occurs when <span class="math">\omega[/spoiler] is zero. which is at the end points.


So we have,
maximum:
occurs at <span class="math">\theta=0[/spoiler], with tension <span class="math">mg+m\omega^{2}r[/spoiler]

minimum:
occurs at <span class="math">\theta=\theta_{max}[/spoiler], with tension <span class="math">mg cos(\theta_{max})[/spoiler]

[Anonymous No.4259744]

>>4259739
sorry, in the value for maximum tension, make that <span class="math">\omega[/spoiler] into <span class="math">\omega_{max}[/spoiler].
Or <span class="math">\omega(0)[/spoiler] (because <span class="math">\omega[/spoiler] is a function of <span class="math">\theta[/spoiler].)