Pendulum Problem

Question

Directions: Complete the following problem showing as much work as possible.

You are considering installing a pendulum in the amusement park visitor’s center. The time it takes a pendulum to complete a full cycle or swing depends upon the length of the pendulum. The formula is given by T = 2 π √(L/32) where T represents the time in seconds and L represents the length of the pendulum in feet.

If the pendulum is 8 feet long, how long will it take for the pendulum to complete a full cycle? (10 points)

You’d like your pendulum to take 20 seconds to complete a cycle. How long will the pendulum need to be? Is this reasonable in your building, which is planned to be 20 feet high? (10 points)

---PLEASE GIVE EXACT ANSWERS PLEASE---

Answer 1

T = 2 π √(8)/(32) = 2 π √(1/4) = 2 π (1/2) = π ∼ 3.14159... seconds

Solving for L:

T = 2 π √(L/32)

Solving for L:

T = ( (2 π) / √(32) ) √(L) = ( (2 π) / 4 √(2) ) √(L) = ( π / (2 √(2)) ) √(L)

√(L) = T / ( π / (2 √(2)) ) = (2 √2 ) T / π

squaring both sides:

L = 8 T² / π² or 8 (T/π)²

for T = 20 seconds:

L = 8 (20/π)² = 324.2 feet which is obviously much greater than 20 feet

Answer 2

T = 2π√L/32 = 2π√8/32 = π s = 3.14 s

L = 32T²/(4π²) = 32·20²/(4π²) = 324.23 ft > 20 ft. Not reasonable