Solving for Maximum Speed of Ball in a Pendulum Swing

In summary, the problem involves a 5 kg ball hanging from a 10 m string and being swung horizontally at a 90 degree angle from its equilibrium position. Using the law of conservation of total mechanical energy, the maximum speed of the ball during its swing can be found by calculating the change in gravitational potential energy. The angular velocity is equal to the square root of g/l, but this is different from the angular frequency which is equal to 2*pi/T. The two are only equal for uniform circular motion, which is not the case for a mathematical pendulum.
  • #1
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A 5 kg ball hangs from a 10 m strong. The ball is swung horizontally outward 90 degrees from its equilibrium position. Assuming the system behaves as a simple pendulum, find the maximum speed of the ball during its swing.


what would I have to do to figure this problem out?

[tex]\omega = \sqrt{\frac{g}{l}}[/tex]
[tex] \omega = \sqrt{\frac{9.8}{10}}[/tex]
[tex] \omega=0.99rad/s[/tex]

[tex]\omega r =v[/tex]
[tex]0.99rad/s* 10m =v[/tex]
[tex]v=9.9m/s[/tex]

I'm not getting the answer of 14, what am I doing wrong?
 
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  • #2
HINT:Use the law of conservation of total mechanical energy.

Daniel.
 
  • #3
yes conservation of energy is always better than mechanics when it comes to fussy math equations. Think of the change in gravitation potential energy.
 
  • #4
And BTW,[itex]v=\omega R [/itex] could work in this case if u knew the maximum angular velocity...

Daniel.
 
  • #5
okay, i used [tex]gh=.5v^2[/tex] and got the answer I was looking for

As for [tex]v=\omega R[/tex], isn't that what [tex]\omega=\sqrt{\frac{g}{l}}[/tex] is? what is omega in that previous equation if it isn't angular velocity?
 
  • #6
Nope,angular velocity is a very complicated function (something involving elliptic functions "cn" and "dn"),because the linear approximation [itex] \sin \vartheta\simeq \vartheta [/itex] would not be correct...

Daniel.
 
  • #7
I read in the book that omega in [tex]\omega=\sqrt{\frac{g}{l}}[/tex] is angular frequency. How is that different from angular velocity?
 
  • #8
Angular velocity is

[tex] \omega (t)=:\frac{d\vartheta (t)}{dt} [/tex]

and angular frequency is

[tex] \omega =:\frac{2\pi}{T} [/tex]

These 2 #-s (denoted the same :yuck:) are equal only for a uniform circular motion .The bob from a mathematical pendulum (not even in the linear approximation) doesn't have a uniform circular motion,ergo the two "animals" are different.


Daniel.
 

1. What is the maximum speed of a ball in a pendulum swing?

The maximum speed of a ball in a pendulum swing is dependent on the length of the pendulum, the angle at which it is released, and the force of gravity. It can be calculated using the equation v = √(2gL(1-cosθ)), where v is the maximum speed, g is the acceleration due to gravity, L is the length of the pendulum, and θ is the angle at which it is released.

2. How does the length of the pendulum affect the maximum speed of the ball?

The length of the pendulum has a direct effect on the maximum speed of the ball. The longer the pendulum, the higher the maximum speed will be. This is because a longer pendulum has a larger arc and therefore a longer distance to travel, resulting in a higher maximum speed.

3. Can the maximum speed of the ball be affected by the angle at which it is released?

Yes, the angle at which the ball is released can affect its maximum speed. The higher the angle, the higher the maximum speed will be. This is because a higher angle results in a larger distance for the ball to travel, resulting in a higher maximum speed.

4. What is the formula for calculating the maximum speed of the ball in a pendulum swing?

The formula for calculating the maximum speed of the ball in a pendulum swing is v = √(2gL(1-cosθ)), where v is the maximum speed, g is the acceleration due to gravity, L is the length of the pendulum, and θ is the angle at which it is released.

5. Can the maximum speed of the ball in a pendulum swing ever exceed the speed of light?

No, the maximum speed of the ball in a pendulum swing can never exceed the speed of light. According to the theory of relativity, the speed of light is the ultimate speed limit in the universe, and nothing can travel faster than it.

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