What is the angle of inclination θ?

Steps to Solve

1. Identify forces acting on the sphere
   The sphere has two main forces acting on it: the tension ($T$) in the string and the gravitational force ($mg$). The tension can be broken into two components: a vertical component ($T_y$) and a horizontal component ($T_x$).

2. Set up the equations for vertical and horizontal components
   In the vertical direction, the tension's vertical component balances the gravitational force: $$ T_y = mg $$
   In the horizontal direction, the tension's horizontal component provides the necessary centripetal force for circular motion: $$ T_x = \frac{mv^2}{r} $$

3. Use trigonometric relationships
   The components of tension can be related to the angle of inclination $\theta$:

• For vertical: $T_y = T \cos(\theta)$
• For horizontal: $T_x = T \sin(\theta)$

4. Combine the equations
   From the vertical component: $$ T \cos(\theta) = mg $$
   From the horizontal component: $$ T \sin(\theta) = \frac{mv^2}{r} $$
   Dividing the second equation by the first gives: $$ \frac{T \sin(\theta)}{T \cos(\theta)} = \frac{mv^2/r}{mg} $$
   This simplifies to: $$ \tan(\theta) = \frac{v^2}{rg} $$

5. Substitute known values
   We know:

• $v = 2.4 , \text{m/s}$
• $r = 0.50 , \text{m}$
• $g \approx 9.81 , \text{m/s}^2$

Now substitute the values into the equation: $$ \tan(\theta) = \frac{(2.4 , \text{m/s})^2}{(0.50 , \text{m})(9.81 , \text{m/s}^2)} $$

6. Calculate $\tan(\theta)$ and find $\theta$
   Calculate the left-hand side: $$ \tan(\theta) = \frac{5.76}{4.905} \approx 1.173 $$
   Now find $\theta$ using the arctangent: $$ \theta = \tan^{-1}(1.173) $$
   Calculating that gives $\theta \approx 50.1^\circ$.