Tetherball - find theta and tension

In summary: These answers line up with the book's answers. Thanks again!In summary, a 450-g tetherball is moving along a horizontal circular path at a constant speed. The cord forms an angle θ with the pole BC, and the tension in the cord is T.
  • #1
JJBladester
Gold Member
286
2

Homework Statement


A 450-g tetherball A is moving along a horizontal circular path at a constant speed of 4m/s. Determine (a) the angle θ that the cord forms with pole BC, (b) the tension in the cord.
tetherball%20illustration.png


Homework Equations


[tex]\sum F_{y}=ma_{y}[/tex]
[tex]\sum F_{n}=ma_{n}=m\frac{v^{2}}{r}[/tex]

The Attempt at a Solution



Find θ and T.

Given:
m=0.45kg
v=4m/s

FBD:
tetherball.jpg


Sum of forces equations:
(1)[tex]\sum F_{y}=-mg+Tsin\theta =0 \to Tsin\theta =mg[/tex]
(2)[tex]\sum F_{n}=Tcos\theta =m\frac{v^{2}}{r} \to T=m\frac{v^{2}}{rcos\theta}[/tex]

Substitute expression for T found in (2) into (1):
[tex]\left (m\frac{v^{2}}{r} \right )tan\theta =mg[/tex]
[tex]\theta =tan^{-1}\left (\frac{gr}{v^{2}} \right )[/tex]

This is where I get stuck. I know that if I can find r, I can find θ and subsequently T. How do I find r?
 
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  • #2
You have found that

tanθ = gr/v^2.

Check this expression. In FBD angles indicated are different.

In the problem, the length of cord L is given. So sinθ = r/L.

Put it in the expression and find θ.
 
  • #3
rl.bhat said:
Check this expression. In FBD angles indicated are different.

I fixed my FBD so that θ is in the correct place:
tetherball%20new.png


[tex]sin\theta=\frac{r}{1.8}\rightarrow r=1.8sin\theta[/tex]

[tex]\sum F_{y}=-mg+Tcos\theta=0[/tex]

(1) [tex]Tcos\theta=mg[/tex]


[tex]\sum F_{n}=Tsin\theta=m\frac{v^{2}}{r}[/tex]

(2) [tex]T=\frac{mv^{2}}{rsin\theta}[/tex]

Substituting (2) into (1) yields:

[tex]\left (\frac{mv^2}{rsin\theta} \right )cos\theta=mg[/tex]

[tex]\frac{cos\theta}{sin^2\theta}=\frac{1.8g}{v^2}[/tex]

How do I simplify the [itex]\frac{cos\theta}{sin^2\theta}[/itex] to solve for θ?
 
Last edited:
  • #4
JJBladester said:
How do I simplify the [itex]\frac{cos\theta}{sin^2\theta}[/itex] to solve for θ?
Hint: Look up the Pythagorean trig identities. (I'm sure it's one you already know.)
 
  • #5
Doc Al said:
Hint: Look up the Pythagorean trig identities...

I did this:
[tex]\frac{cos\theta}{sin^2\theta}=\frac{cos\theta}{1-cos^2\theta}[/tex]

Making the equation:
[tex]\frac{cos\theta}{1-cos^2\theta}=\frac{1.8g}{v^2}[/tex]

However, this doesn't get me any closer to getting theta out on its own.
 
  • #6
JJBladester said:
However, this doesn't get me any closer to getting theta out on its own.
Sure it does. Hint: Rearrange that equation and solve for cosθ. (You'll get a quadratic.)
 
  • #7
Doc Al said:
You'll get a quadratic...

Thanks to both of you for the persistence. With your help, the solution presented itself:

[tex]\left (\frac{1.8g}{v^2} \right )cos\theta+cos\theta-\left (\frac{1.8g}{v^2} \right )=0[/tex]

[tex]cos\theta=\frac{-1+\sqrt{1^2-4\left (\frac{1.8g}{v^2} \right )\left (\frac{-1.8g}{v^2} \right )}}{2\left (\frac{1.8g}{v^2} \right )}[/tex]

[tex]\theta\approx49.9[/tex] degrees

[tex]T=\frac{mg}{cos\theta}=\frac{\left (0.45kg \right )\left (9.81m/s^2 \right )}{cos(49.9)}\approx6.85N[/tex]

These answers line up with the book's answers. Again, thanks to both of you for the help.
 

Related to Tetherball - find theta and tension

1. What is the purpose of finding theta and tension in tetherball?

The purpose of finding theta and tension in tetherball is to understand the physics behind the game. By calculating these values, one can determine the forces acting on the ball and the pole, which can help players strategize and improve their gameplay.

2. How is theta and tension calculated in tetherball?

Theta can be calculated by measuring the angle between the ball's direction of motion and the tether. Tension can be calculated using the equation T=mv^2/r, where T is tension, m is the mass of the ball, v is the velocity of the ball, and r is the radius of the circle that the ball follows.

3. What factors can affect the values of theta and tension in tetherball?

The values of theta and tension can be affected by factors such as the force applied by the player, the length and material of the tether, and the weight and size of the ball. Other factors, such as wind and surface friction, can also affect these values.

4. How can knowing theta and tension improve my tetherball skills?

Knowing theta and tension can help players understand the mechanics of the game and make strategic decisions, such as adjusting their swing angle or applying more force to the ball. It can also help players determine the optimal placement of the ball on the tether for a better shot.

5. Are there any real-world applications of calculating theta and tension in tetherball?

Yes, the principles of calculating theta and tension in tetherball can be applied to other sports and activities, such as swinging a bat or hitting a tennis ball. It can also be useful in engineering and design, as understanding forces and angles is essential in building structures and machines.

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