A 0.50 kg ball that is tied to the end of a 1.1 m light cord is revolved in a horizontal plane with the cord making a 30° angle, with the vertical
(a) Determine the ball's speed.
(b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical?
(a) To determine the ball's speed, we can use the concept of centripetal force. When an object moves in a circle, there is a force acting towards the center, called the centripetal force, that keeps the object in circular motion. It is given by the equation:
F = m * v^2 / r
where F is the centripetal force, m is the mass of the object, v is the velocity (speed) of the object, and r is the radius of the circular path.
In this case, the centripetal force is provided by the tension in the cord, and it can be found by analyzing the forces acting on the ball. There are two forces: the tension force and the weight of the ball. The weight of the ball can be split into two components: mg * sin(θ), which acts horizontally, and mg * cos(θ), which acts vertically.
Since the ball is in equilibrium, the vertical components of the forces must balance each other:
Tension * cos(θ) = mg * cos(θ)
Here, θ is the angle between the cord and the vertical plane. We are given that θ = 30°.
Solving this equation for Tension, we get:
Tension = mg
Now, the centripetal force is equal to the tension in the cord:
F = Tension = mg
Substituting the values of m and g, we get:
F = (0.50 kg) * (9.8 m/s^2) = 4.9 N
The centripetal force can also be written as:
F = m * v^2 / r
Rearranging this equation to solve for v, we get:
v = √(F * r / m)
Substituting the known values, we get:
v = √[(4.9 N) * (1.1 m) / (0.50 kg)]
v ≈ 4.2 m/s
Therefore, the ball's speed is approximately 4.2 m/s.
(b) If the ball is revolved with a speed of 4.0 m/s, we can use similar steps to find the angle θ.
Again, we start with the equation for the centripetal force:
F = m * v^2 / r
Rearranging this equation to solve for r, we get:
r = m * v^2 / F
Substituting the known values, we get:
r = (0.50 kg) * (4.0 m/s)^2 / (4.9 N)
r ≈ 1.63 m
Now, we can use trigonometry to find the angle θ. In a right triangle with the cord as the hypotenuse and r as the adjacent side, we have:
cos(θ) = adj/hyp = r / 1.1 m
θ = arccos(r / 1.1 m) = arccos(1.63 m / 1.1 m)
θ ≈ 51.2°
Therefore, the cord makes an angle of approximately 51.2° with the vertical.