An athlete swings a 4.30 kg ball horizontally on the end of a rope. The ball moves in a circle of radius 0.790 m at an angular speed of 0.590 rev/s.
(a) What is the tangential speed of the ball?
m/s
(b) What is its centripetal acceleration?
m/s2
(c) If the maximum tension the rope can withstand before breaking is 81 N, what is the maximum tangential speed the ball can have?
m/s
A placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 24.0 m/s at an angle of 45.0° to the horizontal.
To find the answers to these questions, we need to use some basic equations related to circular motion.
(a) The tangential speed of an object moving in a circle can be calculated using the formula:
tangential speed = radius x angular speed
Given: radius = 0.790 m and angular speed = 0.590 rev/s
We need to convert the angular speed from revolutions per second to radians per second:
1 revolution = 2π radian
So, angular speed (in radians per second) = 0.590 rev/s x 2π rad/rev
tangential speed = 0.790 m x (0.590 rev/s x 2π rad/rev)
Now, let's calculate the value:
tangential speed = 0.790 m x (0.590 x 2π) m/s
(b) Centripetal acceleration is the acceleration towards the center of the circle. It can be calculated using the formula:
centripetal acceleration = (tangential speed)^2 / radius
Substituting the values, we get:
centripetal acceleration = (tangential speed)^2 / 0.790 m
(c) To find the maximum tangential speed the ball can have without breaking the rope, we need to consider the tension force acting on the rope.
Tension force is related to the centripetal force by the equation:
Tension force = mass x centripetal acceleration
Given: mass = 4.30 kg and maximum tension = 81 N
We can rearrange the equation to solve for the maximum tangential speed:
maximum tangential speed = (maximum tension force) / (mass x centripetal acceleration)
Plug in the values and calculate:
maximum tangential speed = 81 N / (4.30 kg x centripetal acceleration)