College

A trapeze artist swings in simple harmonic motion with a period of 3.8 s.

Calculate the length of the cables supporting the trapeze. (Assume [tex]g = 9.81 \, \text{m/s}^2[/tex].)

Answer :

The length of the cables supporting the trapeze, calculated using the formula for the period of a simple pendulum, is approximately 3.55 meters.

The question involves calculating the length of the cables supporting a trapeze, where the trapeze artist performs with a motion that can be approximated as simple harmonic motion. The given period of the motion is 3.8 seconds, and the acceleration due to gravity (g) is 9.81 m/s2.

To find the length of the cables (L), we use the formula for the period (T) of a simple pendulum, which is a good approximation for our scenario since the swinging motion of a trapeze can be likened to that of a pendulum. The formula is:

T = 2π√(L/g)

Rearranging this formula to solve for L, we obtain:

L = (T2× g) / (4π2)

Substituting the given values:

L = ((3.8)2 × 9.81) / (4π2)

L = 3.55 meters

Therefore, the length of the cables supporting the trapeze is approximately 3.55 meters.

As we know that time period of simple pendulum is given as

T = 2π √L/g

here we know that

T = 3.8 s

now from above equation we know that

T² = 4π² (L/g)

now on rearranging the above equation we will have

L = gT² / 4π²

now plug in all data into it

L = (9.8) (3.8)² / (4) (3.14)²

so the length of the cable must be 3.6 m