Given data:
Mass (m) = 0.1 kg
Length of string (L) = 20 cm = 0.2 m
Angle made by the string with the vertical (θ) = 30 degrees
Speed of mass (v) = 1.5 m/s

Tangential acceleration (at):
Tangential acceleration is the rate at which the speed of an object changes as it moves along a curved path. It is given by the formula:

at = v^2 / r

where v is the speed of the object and r is the radius of the circular path.

Finding the radius (r):
The radius of the circular path can be found using trigonometry. We can use the sine function to relate the angle θ and the radius r:

sin(θ) = r / L

Rearranging the equation, we get:

r = L * sin(θ)

Substituting the given values, we have:

r = 0.2 * sin(30°)
r = 0.1 m

Calculating the tangential acceleration (at):
Now, we can calculate the tangential acceleration using the formula:

at = v^2 / r

Substituting the given values, we have:

at = (1.5)^2 / 0.1
at = 22.5 m/s^2

Therefore, the tangential acceleration of the mass at that instant is 22.5 m/s^2.

Explanation:
- The problem involves a mass being rotated in a vertical circle using a string.
- The mass is moving along a circular path, and at a certain point, the string makes an angle of 30 degrees with the vertical.
- The speed of the mass at that instant is given as 1.5 m/s.
- To find the tangential acceleration, we need to calculate the radius of the circular path.
- We can use trigonometry and the given angle and length of the string to find the radius.
- Once we have the radius, we can use the formula for tangential acceleration to calculate the value.
- The tangential acceleration represents the rate at which the speed of the mass changes as it moves along the curved path.
- In this case, the tangential acceleration is found to be 22.5 m/s^2.