Answer:
(b) T = 50.0 N
(c) t = 1.99 s
(d) ω = 3.16 rad/s
Explanation:
A free body diagram is a diagram showing all the forces acting on a body. In this problem, there are two forces on the ball: weight force mg pulling down and tension force T pulling up at an angle along the string. The ball moves in a circle and experiences a centripetal acceleration towards the center of the circle. By using Newton's second law of motion, which says that the net force on an object is equal to its mass times its acceleration, we can solve for the tension in the string.
(b) Having drawn the free body diagram, sum the forces in the vertical direction.
∑F = ma
T cos θ − mg = 0
T = mg / cos θ
To find the angle θ, use trigonometry.
sin θ = r / L
sin θ = 0.200 m / 1.00 m
θ = 11.54°
Therefore, the tension force is:
T = (5.00 kg) (9.8 m/s²) / cos 11.54°
T = 50.0 N
(c) Sum the forces in the centripetal direction.
∑F = ma
T sin θ = m v²/r
(50.0 N) (0.2) = (5.00 kg) v² / (0.200 m)
v = 0.633 m/s
The period of rotation (the time to make one revolution) is equal to the circumference of the circle divided by the speed.
t = 2πr / v
t = 2π (0.200 m) / (0.633 m/s)
t = 1.99 s
(d) The ball's angular velocity (ω) is equal to its linear velocity (v) divided by the radius (r).
ω = v / r
ω = (0.633 m/s) / (0.200 m)
ω = 3.16 rad/s
