"the period of a pendulum, T (in seconds) varies directly with the square root of the length of the pendulum, L (in centimeters)". Given that a pendulum of length 9cm has a period of 0.63 seconds, find:
a) The period of a pendulum of length 25cm
b) The length of a pendulum whose period is 1.5 seconds,
thanks :)
Given
T seconds =k √L (cm)
where k is an arbitrary constant to be determined.
Also given
0.63=k√9
Therefore we can determine the constant k.
k=0.63/√9=0.21
The relation now reads:
T(L)=0.21√L
where T is period in seconds
and L is length of pendulum in cm.
The problem reduces to:
a) T(25)=?
b) if T(L)=1.5 s, what is L?
Can you take it from here?
T = k√L
so, T/√L is constant.
(a) .63/√9 = T/√25
(b) .63/√9 = 1.5/√L
To solve this problem, we can use the formula for direct variation: T = k√L, where T is the period in seconds, L is the length in centimeters, and k is the constant of variation.
First, let's find the value of k using the given information: a pendulum of length 9cm has a period of 0.63 seconds.
0.63 = k√9
0.63 = 3k
k = 0.63/3
k = 0.21
a) To find the period of a pendulum with a length of 25cm, we can substitute the values into the formula:
T = 0.21√25
T = 0.21 * 5
T = 1.05 seconds
b) To find the length of a pendulum with a period of 1.5 seconds, we can rearrange the formula:
1.5 = 0.21√L
√L = 1.5/0.21
√L ≈ 7.14
Squaring both sides, we get:
L = (7.14)^2
L ≈ 51 centimeters
Therefore, the period of a pendulum with a length of 25cm is 1.05 seconds, and the length of a pendulum with a period of 1.5 seconds is approximately 51 centimeters.
To solve this problem, we can use the direct variation equation:
T = k√L
where T represents the period of the pendulum, L represents the length of the pendulum, and k is the constant of variation.
a) To find the period of a pendulum with a length of 25cm, we can use the given information to find the value of k and then substitute it into the equation.
Given:
T = 0.63 seconds when L = 9cm
First, let's find the value of k. We can rearrange the equation to solve for k:
T = k√L
0.63 = k√9
0.63 = 3k
Now, divide both sides of the equation by 3 to isolate k:
k = 0.63 / 3
k ≈ 0.21
Now that we have the value of k, we can find the new period T when L = 25cm:
T = k√L
T = 0.21√25
T ≈ 0.21 * 5
T ≈ 1.05 seconds
Therefore, the period of a pendulum with a length of 25cm is approximately 1.05 seconds.
b) To find the length of a pendulum with a period of 1.5 seconds, we can rearrange the equation and solve for L:
T = k√L
1.5 = k√L
Now, substitute the value of k we previously found:
1.5 = 0.21√L
Next, square both sides of the equation to eliminate the square root:
(1.5)^2 = (0.21√L)^2
2.25 = 0.0441L
Now, divide both sides of the equation by 0.0441 to solve for L:
L = 2.25 / 0.0441
L ≈ 51.02 cm
Therefore, the length of a pendulum with a period of 1.5 seconds is approximately 51.02 cm.