A 0.50 kg ball that is tied to the end of a 1.4 m light cord is revolved in a horizontal plane with the cord making a 30° angle with the vertical. (b) If the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical?

Duplicate post; already answered

To find the angle that the cord makes with the vertical when the ball is revolved at a speed of 4.0 m/s, we need to use the principles of centripetal force and trigonometry.

First, let's understand the forces acting on the ball when it is in motion. The weight of the ball, which is the force due to gravity acting vertically downwards, can be represented by the equation:

F_weight = m * g

where m is the mass of the ball (0.50 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Next, we have the centripetal force acting towards the center of the circular path that the ball is following. The centripetal force can be calculated using the equation:

F_centripetal = m * v^2 / r

where v is the speed of the ball (4.0 m/s) and r is the radius of the circular path, which is equal to the length of the cord (1.4 m).

In this case, the centripetal force is provided by the tension in the cord. This tension can be split into two components: one vertical and one horizontal. The vertical component cancels out the weight of the ball, while the horizontal component provides the necessary centripetal force.

Since the angle between the cord and the vertical is given as 30°, we can find the vertical component of the tension using trigonometry:

T_vertical = T * sin(30°)

where T is the tension in the cord.

Now, since T_vertical cancels out the weight of the ball, we have:

T_vertical = m * g

Substituting the known values, we can calculate T:

T = (m * g) / sin(30°)

Finally, to find the angle that the cord makes with the vertical, we can use trigonometry once again:

tan(theta) = T_horizontal / T_vertical

where T_horizontal is the horizontal component of the tension in the cord.

Rearranging the equation, we get:

tan(theta) = T_horizontal / (m * g)

Simplifying, we have:

theta = arctan(T_horizontal / (m * g))

Plugging in the values for T_horizontal, m, and g, we can calculate theta.