A tether ball of mass 0.25 kg is attached to a vertical pole by a cord 1.4 m long. Assume the cord attaches to the center of the ball. The cord makes an angle of 20° with the vertical.
(a) What is the tension in the cord?
(b) What is the speed of the ball?
To find the tension in the cord and the speed of the ball, we need to analyze the forces acting on the ball and apply the principles of circular motion.
(a) The tension in the cord can be determined by considering the forces acting on the ball.
The gravitational force (weight) acting on the ball is given by:
F_gravity = m * g
where m is the mass of the ball and g is the acceleration due to gravity.
In this case, m = 0.25 kg and g = 9.8 m/s^2. Therefore:
F_gravity = 0.25 kg * 9.8 m/s^2 = 2.45 N
The tension in the cord can be resolved into two components: one in the vertical direction and one in the horizontal direction.
The vertical component of tension, denoted as F_vertical, balances the weight of the ball:
F_vertical = F_gravity = 2.45 N
The horizontal component of tension, denoted as F_horizontal, provides the centripetal force required to keep the ball moving in a circle. This force can be found using the equation:
F_horizontal = m * v^2 / r
where v is the speed of the ball and r is the radius of the circular path, which is equal to the length of the cord.
To find F_horizontal, we first need to find v. We can use trigonometry to determine the displacement in the horizontal direction, which is equal to r * sin(α), where α is the angle the cord makes with the vertical.
Displacement in the horizontal direction = 1.4 m * sin(20°) = 0.47 m
The time it takes for the ball to complete one full revolution is equal to its period, T. The formula for the period of a pendulum is given by:
T = 2π * sqrt(l / g)
where l is the length of the pendulum and g is the acceleration due to gravity.
In this case, l = 1.4 m. Therefore:
T = 2π * sqrt(1.4 / 9.8) ≈ 3.14 s
Since the ball completes one full revolution in the period T, the speed of the ball, v, can be calculated by:
v = (displacement in the horizontal direction) / T
v = 0.47 m / 3.14 s ≈ 0.15 m/s
Now, we can calculate the horizontal component of tension, F_horizontal:
F_horizontal = m * v^2 / r
F_horizontal = 0.25 kg * (0.15 m/s)^2 / 1.4 m ≈ 0.038 N
Therefore, the tension in the cord is the vector sum of the vertical and horizontal components:
Tension = √(F_vertical^2 + F_horizontal^2)
Tension = √((2.45 N)^2 + (0.038 N)^2)
Tension ≈ 2.45 N
(b) The speed of the ball is approximately 0.15 m/s, as calculated in part (a).