A christmas bulb of mass W is swung in a vertical circle on a cord length Y. What minimum speed v needed at the top of the circle to keep the cord taut?
v^2/Y = g
v = sqrt(Y*g)
W does not matter.
A 55kg Man is on a merry go round and is 15m from the center. It takes 4.5 seconds for 1 complete trip. What is ac and fc?
To find the minimum speed required at the top of the circle to keep the cord taut, we can use the concept of equilibrium of forces.
At the topmost point of the circle, the bulb is at its highest position and the cord is completely vertical. At this point, the only forces acting on the bulb are its weight (mg) and the tension in the cord (T). The tension in the cord provides the necessary centripetal force to keep the bulb moving in a circle.
To keep the cord taut, the tension in the cord must be at least equal to the weight of the bulb. So, we have:
T ≥ mg
Where:
T = tension in the cord
m = mass of the bulb
g = acceleration due to gravity (9.8 m/s^2)
Now, let's consider the forces acting on the bulb at the top of the circle. The tension in the cord provides the centripetal force which is given by:
Fc = m * v^2 / r
Where:
Fc = centripetal force
m = mass of the bulb
v = velocity of the bulb at the top of the circle
r = radius of the circle (equal to the length of the cord, Y)
Since the tension in the cord (T) provides the centripetal force (Fc), we can equate the two:
T = Fc
So, we have:
T = m * v^2 / r
Now, substitute T ≥ mg:
mg ≥ m * v^2 / r
Cancel out the mass 'm':
g ≥ v^2 / r
Rearranging the equation to solve for v:
v^2 ≤ g * r
Taking the square root of both sides:
v ≤ √(g * r)
Therefore, the minimum speed (v) required at the top of the circle to keep the cord taut is given by:
v = √(g * r)
And substituting the given values:
v = √(9.8 * Y)