A string 50cm long and weighing 1.0gram produces a note of 100Hz on plucking. What is the tension in the string?

To find the tension in the string that produces a note of 100 Hz, we can use the formula that relates the frequency of a vibrating string to its physical properties. The formula is:

Understanding the Formula

The frequency (f) of a vibrating string is given by:

f = (1/2L) * √(T/μ)

Where:

• f = frequency in hertz (Hz)
• L = length of the string in meters (m)
• T = tension in the string in newtons (N)
• μ = linear mass density of the string in kilograms per meter (kg/m)

Calculating the Linear Mass Density

Before we can use the formula, we need to determine the linear mass density (μ) of the string. This is calculated as:

μ = mass / length

Given that the mass of the string is 1.0 gram (which is 0.001 kg) and the length is 50 cm (which is 0.5 m), we can substitute these values in:

μ = 0.001 kg / 0.5 m = 0.002 kg/m

Rearranging the Formula for Tension

Now that we have μ, we can rearrange the original frequency formula to solve for tension (T):

T = (2Lf)² * μ

Plugging in the Values

Now we can substitute the values:

• L = 0.5 m
• f = 100 Hz
• μ = 0.002 kg/m

Substituting into the rearranged formula gives us:

T = (2 * 0.5 m * 100 Hz)² * 0.002 kg/m

This simplifies to:

T = (100 m/s)² * 0.002 kg/m

T = 10000 m²/s² * 0.002 kg/m

T = 20 N

Final Result

Thus, the tension in the string is 20 Newtons. This value represents the force needed to maintain the string's vibration at the specified frequency of 100 Hz.