Which of the following equations can be a harmonic on a string that is 10 cm long? Select all that apply. (three correct answers)

A.) y=2sin (pi/5 x)

B.) y=2sin (2pi/7 x)

C.) y=2sin (pi/10 x)

D.) y=2sin (10pi x)

E.) y=2sin (5/2pi x)

The options are: A, C and D

Step-by-step explanation:

The sine wave has a general form : y = A sin (BX)

Where A is the amplitude and B = 2π/period

So, we will check which of the options will be a harmonic on a string that is 10 cm long.

A.) y=2sin (pi/5 x)

B = π/5 ⇒ period = 2π/B = 2π : π/5 = 2π * 5/π = 10

So, one cycle of y=2sin (pi/5 x) will be a harmonic on a string that is 10 cm long.

B.) y=2sin (2pi/7 x)

B = 2π/7 ⇒ period = 2π/B = 2π : 2π/7 = 7

C.) y=2sin (pi/10 x)

B = π/10 ⇒ period = 2π/B = 2π : π/10 = 20 = 2 * 10

So, half a cycle of y=2sin (pi/10 x) will be a harmonic on a string that is 10 cm long.

D.) y=2sin (10pi x)

B = 10π ⇒ period = 2π/B = 2π : 10π = 1/5 = 10/50

So, 50 cycles of y=2sin (10pi x) will be a harmonic on a string that is 10 cm long.

E.) y=2sin (5/2pi x)

B = 5/2π ⇒ period = 2π/B = 2π : (5/2π) = 4π²/5

So, options A, C and D can be a harmonic on a string that is 10 cm long.