Mechanics Pendulum Homework: Solve Horizontal & Vertical Components

In summary, The problem involves a mass rotating around a circle with a certain angle and it can be solved by finding the horizontal and vertical components of the tension force. The horizontal component is found by using the formula T_x=\omega^2lm\sin(\alpha) and the vertical component is found by using the formula T_y=T_x\tan(\alpha). Then, the equations F_{net,y}=0=T_y-mg and T_x=\omega^2lm\sin(\alpha) can be substituted to solve for the unknowns.
  • #1
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Homework Statement


http://img198.imageshack.us/img198/2900/mechanicspendulum1.jpg

Homework Equations



[tex]w=\frac{d\theta}{dt}[/tex] (1)

[tex]v=rw[/tex] (2)

[tex]F=ma[/tex] (3)

[tex]a=\frac{v^2}{r}[/tex] (4)


The Attempt at a Solution



I'm guessing that the horizontal component is using formula (4) where [tex]r=l.sin(\alpha)[/tex] and v is found from formula (2) such that [tex]v=l.sin(\alpha).w[/tex]

Thus, [tex]a=\frac{(l.sin(\alpha).w)^2}{l.sin(\alpha)}=l.sin(\alpha).w^2[/tex]

However, for the vertical component, I'm unsure how to begin. Oh and I'm not certain if I'm solving the horizontal correctly either, so don't hesitate to scold my mistakes :smile:
 
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  • #2
It's not really a pendulum problem as the mass doesn't oscillate through an energy valley. Consider an angle θ such that it lies on the circle around which the mass revolves. It can be said that [tex]\omega=\frac{d\theta}{dt}[/tex].

The net force [tex]F_{net,y}[/tex] on the mass must be zero, and a free body diagram of the mass will note that [tex]F_{net,y}=0=T_{y}-mg[/tex].

For the mass to revolve in a circle, there must be a centripetal force [tex]T_{x}=m\frac{v^{2}}{r}=m\omega^{2}r[/tex]; in this case, [tex]r=l\sin(\alpha)[/tex] and x component of tension can be rewritten as [tex]T_{x}=\omega^{2}lm\sin(\alpha)[/tex].

Noting that [tex]\vec{T}=T_{x}\vec{i}+T_{y}\vec{j}[/tex], [tex]T_{x}=T\sin(\alpha)[/tex] and [tex]T_{y}=T\cos({\alpha})[/tex]. This gives [tex]T_{x}=T_{y}\tan(\alpha)[/tex].

After some substitution, [tex]mg\tan(\alpha)=\omega^{2}lm\sin(\alpha)[/tex] and [tex]\omega^{2}=\frac{g}{l\cos(\alpha)}[/tex]
 
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  • #3
Thanks, the Tx makes a lot of sense now!
But I'm unfamiliar with the notations you've used here and so I couldn't follow it from there-on:

zcd said:
Noting that [tex]\vec{T}=T_{x}\vec{i}+T_{y}\vec{j}[/tex], [tex]T_{x}=T\sin(\alpha)[/tex]
 
  • #4
I just separated it into components to form a right triangle. From the right triangle, you can see how each component is related to the other component and the tension force vector itself.
 
  • #5
Oh ok I see. While I still can't figure out what that vector notation is meant to represent (somehow, a right triangle), I can see how you got [tex]T_x=T_y.tan\alpha[/tex]

Ok but now, what did you substitute and into which equations?
 
  • #6
There's this:

zcd said:
[tex]F_{net,y}=0=T_{y}-mg[/tex].

And this:

[tex]T_{x}=\omega^{2}lm\sin(\alpha)[/tex].
 
  • #7
Aha now it all makes sense! So for most questions like these, to resolve the horizontal and vertical components, I should find the horizontal in terms of [tex]m,\omega,l,\alpha[/tex] and then the vertical in terms of the horizontal tension force.
 

FAQ: Mechanics Pendulum Homework: Solve Horizontal & Vertical Components

1. What is a mechanics pendulum?

A mechanics pendulum is a physical system consisting of a weight attached to a rod or string that is able to swing freely back and forth. It is often used as a model to study the principles of simple harmonic motion and energy conservation.

2. How do you solve for the horizontal and vertical components in a pendulum?

To solve for the horizontal and vertical components in a pendulum, you first need to determine the initial conditions such as the length of the pendulum, the angle of the string or rod, and the mass of the weight. Then, you can use trigonometric functions to calculate the horizontal and vertical components of the weight's displacement at any given time.

3. What is the difference between the horizontal and vertical components in a pendulum?

The horizontal component of a pendulum's displacement represents the side-to-side motion, while the vertical component represents the up-and-down motion. The two components are perpendicular to each other and together they make up the full displacement of the pendulum.

4. How does the length of the pendulum affect its horizontal and vertical components?

The length of the pendulum affects both the horizontal and vertical components in different ways. As the length of the pendulum increases, the amplitude (maximum displacement) of both components decreases, resulting in a slower swing. However, the vertical component is affected more than the horizontal component. This is because the vertical component is directly proportional to the length of the pendulum, while the horizontal component is only affected indirectly through trigonometric functions.

5. How do horizontal and vertical components affect the period of a pendulum?

The horizontal and vertical components do not affect the period of a pendulum. The period, or the time it takes for one complete swing, is only affected by the length of the pendulum and the acceleration due to gravity. This is because the horizontal and vertical components are in constant balance and do not change the overall motion of the pendulum.

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