Understanding the problem
The question asks us to find the phase difference between two points on a vibrating string fixed at both ends. The equation of the string's vibrations is given as y=16sinpix/15cos96pit, where x and y are in cm and t is in seconds.

Identifying the points
We are given two points, x=13cm and x=16cm. Let's call them point A and point B, respectively.

Calculating the phase difference
To calculate the phase difference between point A and point B, we need to find the phase at each point and then subtract them.

The phase of a point on a vibrating string is given by the argument of the sine function in the equation of the string's vibrations. So, the phase at point A (x=13cm) is given by:

phase at A = pix/15

Similarly, the phase at point B (x=16cm) is given by:

phase at B = 96pit - 16pix/15

To get the phase difference between point A and point B, we subtract the phase at point A from the phase at point B:

phase difference = phase at B - phase at A
= 96pit - 16pix/15 - pix/15
= 95pit - pix/15

Simplifying the expression
We can simplify the expression by factoring out pi/15:

phase difference = pi/15(95t - x)

Final answer
The phase difference between point A and point B is pi/15(95t - x) radians.

Explanation
The vibrations of a string fixed at both ends can be represented by a sine function. The argument of the sine function is the phase of the vibrations at a particular point on the string. To calculate the phase difference between two points, we need to find the phase at each point and then subtract them. In this case, we were given the equation of the vibrations and the positions of the two points. We used the equation to find the phases at each point and then subtracted them to get the phase difference. Finally, we simplified the expression to get the final answer in a more compact form.