The type of rubber band used inside some baseballs and golf balls obeys Hooke's law over a wide range of elongation of the band. A segment of this material has an unstretched length l = 1.50 m and a mass m = 5.6 g. When a force F = 17 N is applied, the band stretches an additional length Δl = 0.51 m. (a) What is the speed of transverse waves on this stretched rubber band? (b) Find the time required for a transverse pulse to travel the length of the rubber band.
To solve this problem, we need to use the formula for the speed of transverse waves in a stretched material. The formula is given by:
v = sqrt(T / μ),
where v is the speed of the wave, T is the tension in the string, and μ is the linear mass density of the string (mass per unit length).
(a) To find the speed of transverse waves on the stretched rubber band, we first need to find the tension in the band. We can do this using Hooke's law:
F = k * Δl,
where F is the force applied, k is the spring constant, and Δl is the change in length.
Rearranging the equation, we have:
k = F / Δl.
Now, we can calculate the spring constant:
k = 17 N / 0.51 m = 33.33 N/m.
Next, we need to find the linear mass density (μ) of the rubber band. Linear mass density is defined as the mass per unit length, so we can calculate it using the formula:
μ = m / l,
where m is the mass of the segment and l is the unstretched length.
μ = 5.6 g / 1.50 m = 3.73 g/m = 0.00373 kg/m.
Now, we can substitute the values of T and μ into the formula for the speed of transverse waves:
v = sqrt(T / μ) = sqrt((33.33 N/m) / (0.00373 kg/m)) = 349.26 m/s.
Therefore, the speed of transverse waves on the stretched rubber band is approximately 349.26 m/s.
(b) To find the time required for a transverse pulse to travel the length of the rubber band, we use the formula:
t = L / v,
where t is the time, L is the length of the rubber band, and v is the speed of the wave.
Substituting the values, we get:
t = (1.50 m + 0.51 m) / 349.26 m/s = 0.0060 s.
Therefore, the time required for a transverse pulse to travel the length of the rubber band is approximately 0.0060 seconds.
v=radical(F/m)
m=M/(l+delta (l))
t=l+delta l/ v