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. determine the balls speed.
(b) If the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical?
go gulag
To determine the ball's speed, we can apply the principles of circular motion. We need to find the centripetal force acting on the ball and then use it to calculate the speed.
First, let's draw a diagram to visualize the situation:
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In this diagram, the ball is tied to the end of the cord and is moving in a horizontal plane. The cord makes a 30° angle with the vertical.
To find the centripetal force, we need to resolve the weight of the ball into its horizontal and vertical components. The component along the horizontal plane will provide the necessary centripetal force.
The vertical component of the weight is given by:
Vertical Component = Weight * sin(angle)
= m * g * sin(angle)
= 0.50 kg * 9.8 m/s^2 * sin(30°)
The horizontal component of the weight provides the centripetal force:
Centripetal Force = Horizontal Component of Weight
= m * g * cos(angle)
= 0.50 kg * 9.8 m/s^2 * cos(30°)
The centripetal force is given by:
Centripetal Force = m * v^2 / r
where m is the mass of the ball, v is its velocity, and r is the radius of the circular path.
Since the cord length is given as 1.4 m, the radius of the circular path is half of this value:
r = 1.4 m / 2 = 0.7 m
Now, we can set up the equation:
m * g * cos(30°) = m * v^2 / r
Plugging in the values we know:
0.50 kg * 9.8 m/s^2 * cos(30°) = 0.50 kg * v^2 / 0.7 m
Simplifying the equation:
4.90 m/s^2 * cos(30°) = v^2 / 0.7 m
Now, we can solve for v:
v^2 = (4.90 m/s^2 * cos(30°)) * 0.7 m
v^2 = 3.786 m^2/s^2
v = √(3.786 m^2/s^2)
Calculating the square root:
v ≈ 1.94 m/s
Therefore, the ball's speed is approximately 1.94 m/s.