Animated Solution for Physics - Waves: A transverse wave travels on a taut steel wire with a velocity of $v$ when tension in it is $2.06\times 10^4$ N. When the tension is changed to $T$, the velocity changed to $v/2$. The value of $T$ is close to

Imagine you are holding a taut steel wire. When you pluck it, a transverse wave travels along its length. The speed at which this wave propagates isn't random; it is governed by a beautiful and simple physical law. The speed $v$ of a transverse wave on a stretched string is given by the equation:

Here, $T$ represents the tension in the string (how hard it is being pulled), and $\mu$ represents the linear mass density (how heavy the string is per unit length).

In our specific problem, we are dealing with the exact same steel wire in two different scenarios. Because the physical wire hasn't changed, its mass and length remain constant. This means the linear mass density $\mu$ is a constant value throughout our experiment.

Since $\mu$ is constant, we can establish a direct proportionality between the wave speed and the tension:

This proportionality is the key to unlocking the problem. It tells us that any change in the wave speed is solely due to a change in the tension, and they are linked by a square root relationship.

To compare the two states of the wire, we can set up a ratio. Let's call the initial state 1 and the final state 2. The ratio of their speeds will be equal to the square root of the ratio of their tensions:

Now, let's carefully substitute the values given to us: - Initial speed, $v_1=v$ - Final speed, $v_2=\frac{v}{2}$ - Initial tension, $T_1=2.06\times 10^4\text{ N}$ - Final tension, $T_2=T$ (this is what we want to find)

Plugging these into our ratio equation yields:

The $v$ terms on the left side cancel out perfectly, simplifying the equation to:

To eliminate the square root, we must square both sides of the equation. This is a crucial step where many students make silly mistakes. Squaring both sides gives:

Now, it's just a matter of simple algebra to isolate $T$:

Adjusting the decimal point for standard scientific notation, we arrive at our final answer:

This result highlights a profound physical truth: because of the square root relationship, if you want to halve the speed of a wave on a string, you must reduce the tension to one-fourth of its original value. You cannot simply halve the tension!