A string of mass 0.0003 kg/m and 0.5m long is vibrating 200 cycles per second. What must be the tension? What hanging mass would produce the tension?
frequency*wavelength=speed=sqrt(tension/mass/length)
200*1=sqrt(tension/.0003)
square both sides
solve for tension
40000*.0003=tension=hangingmass*g
To find the tension in the string, we can use the equation:
Tension = (mass per unit length) * (velocity of the wave)^2.
Given:
Mass per unit length (m/L) = 0.0003 kg/m.
Velocity of the wave (v) = frequency (f) * wavelength (λ).
Frequency (f) = 200 Hz.
Wavelength (λ) = 2 * length (L) = 2 * 0.5 m = 1 m.
Let's calculate the velocity of the wave:
v = f * λ
= 200 Hz * 1 m
= 200 m/s.
Now, we can calculate the tension:
Tension = (mass per unit length) * (velocity of the wave)^2
= (0.0003 kg/m) * (200 m/s)^2
= 0.0003 kg/m * 40000 m^2/s^2
= 12 N.
Therefore, the tension in the string is 12 N.
To find the hanging mass that would produce this tension, we can use the equation:
Tension = (mass of hanging object) * (acceleration due to gravity).
Given:
Tension = 12 N.
Acceleration due to gravity (g) = 9.8 m/s^2.
Let's calculate the mass of the hanging object:
Tension = (mass of hanging object) * (acceleration due to gravity)
12 N = (mass of hanging object) * 9.8 m/s^2
Rearranging the equation:
mass of hanging object = Tension / acceleration due to gravity
= 12 N / 9.8 m/s^2
= 1.22 kg.
Therefore, the hanging mass that would produce the tension is approximately 1.22 kg.
To find the tension in the string, we can use the formula for the tension in a vibrating string:
Tension (T) = (mass per unit length) * (velocity of the wave)^2
To find the velocity of the wave, we can use the formula:
Velocity of the wave (v) = frequency * wavelength
Since we are given the frequency (200 cycles per second), we need to find the wavelength. The wavelength (λ) can be calculated using the formula:
Wavelength (λ) = 2 * length
Given that the length of the string is 0.5 meters, the wavelength is:
λ = 2 * 0.5 = 1 meter
Now, we can find the velocity of the wave:
v = frequency * wavelength
= 200 * 1
= 200 m/s
Next, we can calculate the tension in the string:
Tension (T) = (mass per unit length) * (velocity of the wave)^2
= 0.0003 kg/m * (200 m/s)^2
= 0.0003 kg/m * 40000 m^2/s^2
= 12 N
Therefore, the tension in the string is 12 Newtons.
To find the hanging mass that would produce this tension, we can use the formula:
Tension (T) = (mass of hanging mass) * (acceleration due to gravity)
Rearranging the equation, we have:
mass of hanging mass = Tension / acceleration due to gravity
Given that the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the mass of the hanging mass:
mass of hanging mass = 12 N / 9.8 m/s^2
= 1.22 kg
So, a hanging mass of approximately 1.22 kg would produce the tension in the string.