Solve Ball on a String for Velocity w/ Mass, Length & Gravity

In summary, the ball swings up from the nail, but then falls back down and hits it. To find V in terms of M L and g, you use the following equations: v^2 = gl cos(angle) and v = sqrt(3.24Lg). Finally, by using conservation of energy, you arrive at V = Sqrt(3.24Lg).
  • #1
melancholy2
4
0
This isn't actually a homework problem, but I thought it would be relevant to post it here since it is a problem afterall, and I need some help at the solution. Thank you everyone!

Homework Statement



There is a ball mass M hanging on a massless string length L, from a nail in the wall. When the ball is projected with initial horizontal velocity V, it swings up, but then falls back down and hits the nail. Find V in terms of M L and g.

2. Requested Help

I would appreciate it if someone could see if my method is problematic, and if my answer is correct if you would want to work it out.

The Attempt at a Solution



For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. Thereafter, the parabolic motion will coincide with the nail. I have 3 equations and 3 unknowns - velocity at moment of transition, time taken for parabolic flight, and angle from vertical which transition occurs.

Using the above, I got an angle of 65.53 deg
I got v^2 = gl cos(angle)
And I eliminated t.

Finally, after using conservation of energy, I arrive at projection velocity V = Sqrt(3.24Lg)
 
Physics news on Phys.org
  • #2
Welcome to PF!

Hi melancholy2! Welcome to PF! :smile:

(have a square-root: √ and a degree: º and try using the X2 tag just above the Reply box :wink:)
melancholy2 said:
For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. …

Yes, that's correct. :smile: But it's very difficult to check your final result without seeing your full calculations. :redface:
 
  • #3
Hi everyone,

I've relooked at my workings and managed to come up with a new answer which I think is correct. Angle is now 54.7 degrees. Please see the attachment solution. Thanks!
 

Attachments

  • solutionball.gif
    solutionball.gif
    29 KB · Views: 429
  • #4
Hi melancholy2! :smile:

(I haven't actually checked the last 6 lines :redface:, but apart from that …)

Yes, that looks fine. :smile:

(except you could have saved yourself a little trouble if you'd noticed that lcosθ + lsin2θ/cosθ = l/cosθ :wink:)
 

1. How do you calculate velocity using mass, length, and gravity in a ball on a string?

The formula for calculating velocity in a ball on a string is v = √(g * L * (1 - cosθ)), where v is velocity, g is the acceleration due to gravity (9.8 m/s²), L is the length of the string, and θ is the angle at which the ball is released from the vertical position.

2. What is the relationship between mass and velocity in a ball on a string?

In a ball on a string experiment, the mass of the ball does not affect the velocity. This is because the formula for velocity (v = √(g * L * (1 - cosθ))) does not include mass as a variable. Therefore, the velocity will remain constant regardless of the mass of the ball.

3. How does the length of the string affect the velocity of a ball on a string?

The length of the string has a direct effect on the velocity of a ball on a string. According to the formula (v = √(g * L * (1 - cosθ))), as the length of the string increases, the velocity will also increase. This is because a longer string will allow the ball to travel a greater distance, resulting in a higher velocity.

4. What is the role of gravity in calculating velocity in a ball on a string?

Gravity plays a crucial role in calculating the velocity of a ball on a string. The acceleration due to gravity (g) is a constant factor in the formula (v = √(g * L * (1 - cosθ))), and it is responsible for the downward force acting on the ball. As the angle at which the ball is released changes, the effect of gravity on the ball also changes, resulting in a change in velocity.

5. Can you determine the velocity of a ball on a string without knowing the length of the string?

No, the length of the string is a crucial component in calculating the velocity of a ball on a string. Without knowing the length of the string, it is impossible to accurately determine the velocity using the formula (v = √(g * L * (1 - cosθ))). Therefore, it is essential to measure the length of the string in order to calculate the velocity of the ball accurately.

Similar threads

Relative velocity passing a ball between two soccer players
    • Introductory Physics Homework Help
Replies
5
Views
2K
Angular momentum conservation in collision with a nail
    • Introductory Physics Homework Help
Replies
9
Views
1K
(Magnetism) Magnetic pull force on a steel ball
    • Introductory Physics Homework Help
Replies
4
Views
355
Foucault Pendulum at the North Pole
    • Introductory Physics Homework Help
Replies
9
Views
704
Deriving Pith Ball Mass from Coulomb's Law: A Scientific Exploration
    • Introductory Physics Homework Help
Replies
7
Views
1K
Finding the frequency of a string based on Mass and Tension
    • Introductory Physics Homework Help
Replies
1
Views
1K
What will the wave velocity be in this string?
    • Introductory Physics Homework Help
Replies
15
Views
1K
Trajectory of a mass element in a string overhanging a pulley
    • Introductory Physics Homework Help
Replies
19
Views
1K
Tension in string when ball is at the top of a circle
    • Introductory Physics Homework Help
Replies
7
Views
8K
Deriving a Formula for Velocity of Ball 2 Based on Mass and Angle of Inclination
    • Introductory Physics Homework Help
Replies
15
Views
1K

Hot Threads

Recent Insights

Back
Top