The length of time that it takes for a pendulum to make one complete swing depends on the length of the pendulum. The time in seconds, T, is related to the length in meters, L, through the equation:
T=2π√((L)/(9.8)) [it is all under square root]
Q1: write in function notation
Q2: state the domain of this function
Q3: Graph the function pendulums up to 20 m in length
Q4: when T = 10, how long would the pendulum be
Q5: If the pendulum was twice as heavy, what would T be?
T(L) = 2π √(L/9.8)
domain: any positive value of L
graph: you'll have to do this, by making a table of values, pick any L, use your calculator to find T
plot all your (L,T) s
when T = 10
10 = 2π √(L/9.8)
square both sides
100 = 4π^2 (L/9.8)
980/(4π^2) = L = 24.824 m
The weight of the pendulum is not a factor, T would be the same.
A1: In function notation, we can represent the relationship between the length of the pendulum and the time it takes for one complete swing as follows:
T(L) = 2π√(L/9.8)
A2: The domain of this function represents the permissible values for the length of the pendulum, which in this case will be positive real numbers. Since the length cannot be negative or zero, the domain of the function is L > 0.
A3: To graph the function, we will assign values for the length of the pendulum (L) and calculate the corresponding time (T). We will take pendulum lengths up to 20 meters. Here is a table of values:
L | T
--------------
1 | 2π√(1/9.8)
2 | 2π√(2/9.8)
3 | 2π√(3/9.8)
...
20 | 2π√(20/9.8)
Plotting these points on a graph will give us the graph of the function, where L is on the x-axis, and T is on the y-axis.
A4: Given that T = 10, we can find the corresponding length of the pendulum, L, by rearranging the equation for T(L) and solving for L:
10 = 2π√(L/9.8)
Squaring both sides: 100 = 4π^2(L/9.8)
Simplifying: L = (100 * 9.8) / (4π^2)
Thus, the length of the pendulum when T = 10 would be approximately L ≈ 24.88 meters.
A5: If the pendulum was twice as heavy, it would not affect the period of the pendulum swing. The formula for the period, T, does not depend on the mass of the pendulum. So, if the pendulum is twice as heavy, the value T would remain the same.