On planet e a pendulum has a length of 4L has a period of 6T, on planet F a pendulum has a length of 3L has a period of 2T what is the ratio of gravitational acceleration of these two planets
since T = 2π√(L/g), g = (4π^2L)/T^2
the ratio of
g1/g2 = (4π^2L1)/(T1^2) / (4π^2L2)/T2^2
= (4L)/(6T)^2 / (3L)/(2T)^2
= (4/36) / (3/4)
= 1/9 * 4/3
= 4/27
g=4π^2L/T^2
So,
gF/gE=(4π^2L/T^2)/(4π^2L/T^2)=(4π^2(3L)/(2T)^2)/(4π^2(4L)/(6T)^2)
gF/gE=(4π^2(3L)/(2T)^2)/(4π^2(4L)/(6T)^2)=(3L/4T^2)/(4L/36T^2)
gF/gE=(3L/4T^2)/(4L/36T^2).
gF/gE=(3/4)*9=27/4=6.75
To find the ratio of gravitational acceleration between two planets, we can use the formula for the period of a pendulum:
T = 2π√(L/g),
where T is the period, L is the length of the pendulum, and g is the gravitational acceleration.
Let's start by finding the value of g for planet E. We know that a pendulum with a length of 4L has a period of 6T. Therefore, we can write the equation as:
6T = 2π√(4L/g).
To simplify, we can square both sides of the equation:
36T^2 = 16π^2(L/g).
Now, let's find the value of g for planet F. Using the given information that a pendulum with a length of 3L has a period of 2T, we can write the equation as:
2T = 2π√(3L/g).
Again, squaring both sides of the equation:
4T^2 = 12π^2(L/g).
Now, let's compare the ratios of the two gravitational accelerations. We can divide the equation for planet E by the equation for planet F:
(36T^2)/(4T^2) = (16π^2(L/g))/(12π^2(L/g)).
Simplifying further:
9 = (16π^2(L/g))/(12π^2(L/g)).
Canceling out the π^2(L/g) terms:
9 = (16/12).
Simplifying fractions:
9 = 4/3.
Therefore, the ratio of gravitational acceleration between planet E and planet F is 4/3.