I don't understand the derivation of the wave equation

[Clara Chung]

If there is a net force along the y-axis, i.e. T sin(θ₂) - T sin(θ₁)
Why is it equals to ma, where a is the acceleration of the piece of string along the y-axis? Shouldn't there be a torque so the piece of string rotates?
Sorry for sounding stupid.

 

[phoenix95]

Shouldn't there be a torque so the piece of string rotates?

I take it that you think of resultant force as a torque, because the two forces appear to act as a couple. By definition A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment. In this case the forces are equal(considering the infinitesimal element), are displaced by (almost)perpendicular distance but not oppositely directed. Hence the resultant is a force along some particular direction.

Now the question is: what is this resultant force? And that is given by:

there is a net force along the y-axis, i.e. T sin(θ2) - T sin(θ1)

I hope that you understand why there's no net force along the x axis.

 

[Chestermiller]

The angle is supposed to be very small so that the sine of the angle is equal to dy/dx. The force balance is in the y direction, and includes the translational inertia. The bending rigidity of the string is considered negligible. What do you estimate for the torque on each section of string?

 

[Clara Chung]

The angle is supposed to be very small so that the sine of the angle is equal to dy/dx. The force balance is in the y direction, and includes the translational inertia. The bending rigidity of the string is considered negligible. What do you estimate for the torque on each section of string?

I don't know how to estimate the torque. I just think that there is a torque by intuition.

 

[Chestermiller]

I don't know how to estimate the torque. I just think that there is a torque by intuition.

The string is not a rigid body, so it doesn’t have to satisfy a moment balance.

 

[Chestermiller]

The string is not a rigid body, so it doesn’t have to satisfy a moment balance.

Besides, for any short section of string, if you take moments of the tensile forces st its ends about its center of mass, they cancel.

 

[Stephen Tashi]

View attachment 218472
Why is it equals to ma, where a is the acceleration of the piece of string along the y-axis?

What do mean by "it"? What do you want "it" to be besides the acceleration of the center of mass of the piece of string along the y-axis? Are you asking about the acceleration vector for the center of mass of the piece of string? - and asking why its only nonzero component is along the y-axis?

 

[Mister T]

If there is a net force along the y-axis, i.e. T sin(θ₂) - T sin(θ₁)
Why is it equals to ma, where a is the acceleration of the piece of string along the y-axis?

That follows directly from Newton's Second Law, $\vec{F}_{net}=m \vec{a}$. The vertical components of those vectors must also be equal: $F_{net,y}=ma_y$.

Shouldn't there be a torque so the piece of string rotates?

Whether there is or not has no bearing on the above.