The number of acres Problem

Question

(a) From its resting position, how long does it take the pendulum to swing one direction, then the other, and then return to its resting position?
(b) What is the pendulum’s maximum displacement?
(c) What percent of a full cycle is completed within 1 s?
(d) How is the pendulum moving during the time period from x = 5 to x = 5.625?
(e) What is the equation of the graph?

Answer 1

What has this to do with "The number of acres Problem" ?

 

(a)

 

That is the period of the pendulum.

 

T = period

 

T = 2π√(L/g)

 

L = length of pendulum (which you did not provide in the problem)

g = acceleration of gravity = 9.8 m/s²

 

Without a value for L, I cannot give you a value for T.

 

(b)

 

It's maximum angular displacement is out to the same angle as its starting position from rest (which you did not give either), but in the opposite direction.

 

(c)

 

% of 1 cycle is (100%)(1s/T)

 

(d)  

 

Evaluate θ(t) at t = 5s and t = 5.625 s to find the angle swept out at those times.  Starting from rest at some initial small angle θ₀, the angle θ swept as a function of time t can be modeled as

 

θ(t) = θ₀cos[(2π/T)t]

 

After you know L, get T, and then plug the values of t in θ(t).

 

(e)

 

See the equation in part (d).