Estimate the force a person must exert on a string attached to a 0.160 kg ball to make the ball revolve in a circle when the length of the string is 0.600 m. The ball makes 1.80 revolutions per second. Do not ignore the weight of the ball. In particular, find the magnitude of FT, and the angle ϕ it makes with the horizontal. [Hint: Set the horizontal component of FT equal to maR; also, since there is no vertical motion, what can you say about the vertical component of FT?]
I have the force but how do you find the angle?
Ac = v^2/r
r = L cos phi
Horizontal:
FT cos phi = m Ac = m v^2/(L cos phi )
Vertical
FT sin phi = m g
solve this first for
FT = m g /sin phi
then
(m g /sin phi)cos phi = m v^2/(L cos phi)
note: m does not matter in the end.
now get v from circumference
2 pi r = v /1.8
v = 1.8*2*pi * L cos phi
v^2 = (1.8*2*pi * L)^2 cos^2 phi
so
( g /sin phi)cos phi = (1.8*2*pi * L)^2 cos^2 phi/(L cos phi)
note cos phi cancels
solve for sin phi and phi
go back and get FT
To find the angle ϕ that the force makes with the horizontal, you can use trigonometry.
First, let's break down the force FT into its horizontal and vertical components. Since there is no vertical motion, the vertical component of FT is equal to the weight of the ball, which can be calculated as Fg = m * g, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Next, let's calculate the horizontal component of FT. We know that the horizontal component of FT is equal to maR, where m is the mass of the ball, a is the acceleration, and R is the radius of the circular motion. The acceleration can be calculated as a = (2πf)^2 * R, where f is the frequency or number of revolutions per second.
Now, you need to find the magnitude of FT. Since FT is the hypotenuse of a right-angled triangle formed by the horizontal and vertical components, you can use the Pythagorean theorem to find the magnitude:
FT^2 = (Fh^2) + (Fv^2)
Where FT is the magnitude of the force, Fh is the horizontal component, and Fv is the vertical component.
Once you have calculated the magnitude of FT, you can find the angle ϕ using the inverse tangent function:
ϕ = arctan(Fv / Fh)
Finally, substitute the values you have obtained into the equations and solve for FT and ϕ.
Note: Make sure to use consistent units throughout the calculations (e.g., meters for length, kilograms for mass, and newtons for force).