We can see here that the maximum angle the string will make with the vertical is approximately 3.31°.
To find the maximum angle the string will make with the vertical, we can use the concept of conservation of energy.
When the ball is at its maximum angle, all of its initial kinetic energy will be converted into potential energy. At this point, the tension in the string will provide the centripetal force necessary to keep the ball moving in a circular path.
Using the given information, we can start by calculating the potential energy of the ball when it reaches its maximum angle. The potential energy (PE) is given by the equation:
PE = m × g × h
Where:
- m is the mass of the ball,
- g is the acceleration due to gravity, and
- h is the height above the reference point (in this case, the lowest point of the swing).
Since the ball is initially at rest and then given an initial velocity of 1.1 m/s, we can assume the initial kinetic energy (KE) is zero.
At the maximum angle, the potential energy will be equal to the initial kinetic energy:
PE = KE
m × g × h = 0.5 × m × v²
where v is the initial velocity of the ball.
Simplifying the equation, we find:
g × h = 0.5 × v²
Now, we can solve for the height h:
h = (0.5 × v²) / g
Substituting the given values, we have:
h = (0.5 × 1.1²) / 9.8
h ≈ 0.06 m
Since the string is 1.1 m long, the maximum angle the string will make with the vertical can be found using trigonometry. The maximum angle θ can be calculated as:
θ = sin⁻¹(h / L)
where L is the length of the string.
Substituting the values, we get:
θ = sin⁻¹(0.06 / 1.1)
θ ≈ 3.31°
Therefore, the maximum angle the string will make with the vertical is approximately 3.31 degrees.