Physics · Waves · NEET
No. Ordinary density (rho) is mass per unit VOLUME, unit kg/m^3. Linear mass density (mu) is mass per unit LENGTH, unit kg/m. They are linked: for a wire of cross-section area A, mu = rho x A. So a wire made of a dense metal (large rho) or a thick wire (large A) both give a large mu. In Waves problems you almost always need mu, not rho, because a string is treated as a 1-D line.
SI unit is kilogram per metre (kg/m or kg m^-1). Dimensional formula is [M L^-1 T^0], i.e. mass over length. In numericals always convert grams to kg and cm to m first, or your mu (and hence wave speed) will be off by a power of ten.
Use mu = rho x A, where A is the cross-section area. For a circular wire of radius r, A = pi r^2, so mu = rho x pi r^2. Example: rho = 8000 kg/m^3, r = 0.5 mm = 0.5 x 10^-3 m gives A = pi x (0.5e-3)^2 = 7.85 x 10^-7 m^2, so mu = 8000 x 7.85e-7 = 6.3 x 10^-3 kg/m.
Wave speed depends on two things: the restoring force (tension T, which pulls the string back) and the inertia (mu, which resists motion). More inertia per metre means each bit of string is harder to move, so the wave travels slower. This gives v = sqrt(T/mu): speed rises with T and falls as mu rises.
If you CUT a uniform string, each piece keeps the same mu, because mass and length shrink in the same ratio (mu is a per-metre property). If you STRETCH the same string (increase its length while mass is fixed), mu actually decreases, because the same mass now spreads over more length. NEET problems usually assume mu is given and constant unless stretching is stated.
If the tension in a stretched string is doubled, the ratio of the initial speed to the final speed of a transverse wave along the string is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
mu = mass / length = m / L. If the wire's material density rho and cross-section area A are known, then mu = rho x A, and for a circular wire mu = rho x pi r^2.
Kilogram per metre (kg/m). Always convert mass to kg and length to m before dividing, otherwise the wave speed comes out wrong.
Mainly in the transverse wave speed formula v = sqrt(T/mu), and in string/sonometer problems for fundamental frequency and harmonics. mu is the inertia term in that formula.
mu is mass per unit length (kg/m) for a 1-D string; rho is mass per unit volume (kg/m^3) for a 3-D body. They are related by mu = rho x A, where A is the cross-section area.
Same material means same rho, but mu = rho x A. A thicker string has a larger cross-section area A, so it has a larger mu even though the material is identical, and waves travel slower on it under the same tension.
