Respuesta :
Answer:
Approximately [tex]2.83\; {\rm s}[/tex].
Explanation:
If a pendulum is swinging at a very small angle, the period [tex]T[/tex] of the pendulum would be approximately proportional to the square root of the length of the pendulum:
[tex]\displaystyle T \approx 2\, \pi\, \sqrt{\frac{L}{g}}[/tex],
Where:
- [tex]L[/tex] is the length of the pendulum, and
- [tex]g[/tex] is the gravitational field strength.
In this question, the length of the pendulum is doubled. Replace [tex]L[/tex] with [tex]2\, L[/tex] to obtain:
[tex]\displaystyle T \approx 2\, \pi\, \sqrt{\frac{2\, L}{g}} = \sqrt{2}\, \left(2\, \pi\, \sqrt{\frac{L}{g}}\right)[/tex].
In other words, the new period of the pendulum would be approximately [tex]\sqrt{2}[/tex] times the original value. Given that the original period of the pendulum is [tex]2.00\; {\rm s}[/tex], the new period would be approximately:
[tex]2.00\, \sqrt{2} \approx 2.83\; {\rm s}[/tex].
Final answer:
When the length of a pendulum is doubled, the new period is approximately 1.414 times the original period. For a pendulum with initial period of 2.00 seconds, doubling its length to 2.0 meters will result in a new period of approximately 2.83 seconds.
Explanation:
The student asked what would be the period of a pendulum if its length is doubled from 1.0 meter to 2.0 meters while maintaining its mass of 2.5 kg. The period of a simple pendulum is given by the formula T = 2π √(L/g), where T is the period in seconds, L is the length in meters, and g is the acceleration due to gravity in m/s² which is approximately 9.8 m/s² on Earth. Since the period depends only on the length of the pendulum and the acceleration due to gravity, and is independent of the mass, we can calculate the new period by taking the square root of the ratio of the new length to the original length.
When the length of the pendulum is doubled, the ratio becomes 2.0/1.0 = 2. Taking the square root of 2 gives us approximately 1.414. Hence, the new period T' is 1.414 times the original period. Since the original period T is 2.00 seconds, the new period T' will be 2.00 seconds * 1.414 ≈ 2.83 seconds.
