In: Physics
If you pluck a string under tension, a transverse wave moves in the positive x-direction, as shown in Figure below. The small mass element oscillates perpendicular to the wave motion as a result of the restoring force provided by the string and does not move in the x-direction. The tension FT (this will be denoted by F in the following explanation) in the string, which acts in the positive and negative x-direction, is approximately constant and is independent of position and time.
The speed of a pulse or wave on a string under tension can be found with the equation
v =
where F is the tension in the string and μ is the mass per length of the string.
μ = m/L
Here, m is the mass of the spring and L is the length of the original length of the unstretched spring.
Write the equation of the force, ie, Tension in spring.
F1 = k(x1 - L)
Write the expression for speed
v1 =
v1 =
v12= k L (x1 - L) / m ..... (a)
When propagate speed (v2)
F2 = k(x2 - L)
Write the expression for speed
v2 =
v2 =
v22 = k L (x2 - L) / m ..... (b)
Divide equation (b) by equation (a)
v22 / v12 = (kL(x2 - L)/m) / (kL(x1 - L)/m)
v22 / v12 = (x2 - L) / (x1 - L)
Substituting the values given in question,
L = 30 cm = 0.3 m;
x1 = 60 cm = 0.6 m ; v1 = 4.5 m/s
x2 = 90 cm = 0.9 m ; v2 = ?
v22 / 4.52 = (0.9 - 0.3) / (0.6 - 0.3)
v22 = 4.52 x (0.6 / 0.3) = 4.52 x 2 = 40.5
v2 = 6.36 m/s
The waves will travel in this spring at a speed of 6.36 m/s if it is stretched to 90 cm.