To solve this problem, we can use the principles of circular motion. Let's break it down step by step:
Step 1: Calculate the velocity of the bob.
The circumference of the circular path is given as 0.75 m. So, the distance traveled by the bob in one complete revolution is equal to the circumference, which can be calculated using the formula:
C = 2Ï€r
Here, r represents the radius of the circular path. Since the radius is not given directly, we need to find it using the length of the string, L.
The length of the string (L) forms the slant height of the cone (which is the hypotenuse of a right-angled triangle) and the radius (r) of the circular path forms the base of the triangle. The vertical height (h) of the triangle can be calculated using the Pythagorean theorem:
h² = L² - r²
Rearranging the equation, we find:
r = √(L² - h²)
Since the bob moves in a circular path, the vertical height (h) is given by:
h = L - R
Where R represents the height of the cone formed by the string.
Now, substituting the known values:
r = √(1.2² - (1.2 - R)²)
r = √(1.44 - 1.44 + 2.4R - R²)
r = √(2.4R - R²)
The circumference is given as 0.75 m, so:
0.75 = 2Ï€r
Substituting the value of r:
0.75 = 2π√(2.4R - R²)
Now, we can solve this equation to find the value of R, which is the height of the cone.
Step 2: Calculate the tension in the string.
The force of tension in the string provides the necessary centripetal force to keep the bob moving in a circular path. At any point on the circular path, the tension can be calculated using the formula:
T = (m × v²) / r
Here, m represents the mass of the bob (0.038 kg), v represents the velocity of the bob, and r represents the radius of the circular path.
Step 3: Calculate the period of the motion.
The period of the circular motion is the time taken for the bob to complete one full revolution. It can be calculated using the formula:
T = (2Ï€r) / v
We now have all the information required to solve for (a) the tension in the string and (b) the period of the motion.