A child is twirling a 0.0120-kg plastic ball on a string in a horizontal circle whose radius is 0.100 m. The ball travels once around the circle in 0.500 s.
(a)
Determine the centripetal force acting on the ball.
(b)
If the speed is doubled, does the centripetal force double? If not, by what factor does the centripetal force increase?
To determine the centripetal force acting on the ball, we can start by calculating the velocity of the ball using the formula:
(v = 2πr / T)
Where:
v = velocity
r = radius
T = time
Given:
Radius (r) = 0.100 m
Time (T) = 0.500 s
Plugging in the values, we get:
v = 2π(0.100) / 0.500
v = 0.400π m/s
Next, we can calculate the centripetal force using the formula:
Fc = (m * v^2) / r
Where:
Fc = centripetal force
m = mass
v = velocity
r = radius
Given:
Mass (m) = 0.0120 kg
Velocity (v) = 0.400π m/s
Radius (r) = 0.100 m
Plugging in the values, we get:
Fc = (0.0120 * (0.400π)^2) / 0.100
Fc = (0.0120 * 0.400^2 * π^2) / 0.100
Fc = 0.000192π^2 N
Therefore, the centripetal force acting on the ball is 0.000192π^2 N.
To answer part (b):
If the speed is doubled, the centripetal force does not double. The centripetal force is proportional to the square of the velocity, so if the speed is doubled, the centripetal force will increase by a factor of 4.
To determine the centripetal force acting on the ball, we can use the following equation:
F = (m * v^2) / r
where:
F is the centripetal force,
m is the mass of the ball,
v is the velocity of the ball,
and r is the radius of the circle.
(a) Given:
m = 0.0120 kg (mass of the ball)
r = 0.100 m (radius of the circle)
v = distance / time
Since the ball travels once around the circle in 0.500 s, we can calculate the velocity using the formula v = distance / time.
The distance travelled by the ball is equal to the circumference of the circle:
distance = 2πr = 2 * 3.14159 * 0.100 m = 0.62832 m
Using the formula v = distance / time:
v = 0.62832 m / 0.500 s = 1.25664 m/s
Now, we can substitute the values into the equation to calculate the centripetal force:
F = (0.0120 kg * (1.25664 m/s)^2) / 0.100 m = 0.0191 N
So, the centripetal force acting on the ball is approximately 0.0191 Newtons.
(b) If the speed is doubled, we need to find out if the centripetal force doubles or increases by a different factor.
Let's consider the formula for centripetal force:
F = (m * v^2) / r
If we double the speed (v), the equation becomes:
F' = (m * (2v)^2) / r
Simplifying the equation:
F' = (m * 4v^2) / r
As you can see, when we double the speed (v), the centripetal force (F') increases by a factor of 4.
Therefore, if the speed is doubled, the centripetal force increases by a factor of 4.
omega = radians/s = 2 pi/.5 = 4 pi
Ac = omega^2 R = (4 pi)^2 * .1
F = m Ac = .012 (4 pi)^2 (.1)
increases as omega^2 so 2^2 = 4 times