A puck of mass m = 1.5 kg slides in a circle of radius r = 20 cm on a frictionless table while attached to a hanging cylinder of mass M = 2.5 kg by a cord through a hole in the table. What is the speed of mass m that keeps the cylinder at rest?
Mg =mv^2/r
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To find the speed of mass m that keeps the cylinder at rest, we can start by analyzing the forces acting on the system.
At any point in the circular path, there are two forces acting on mass m: the tension in the cord (T) and the centrifugal force (F). The tension provides the centripetal force needed to keep mass m moving in a circle, while the centrifugal force is the apparent outward force experienced by mass m due to the circular motion.
The tension in the cord is responsible for providing the centripetal force. Since there is no friction and the table is frictionless, the only other horizontal force acting on mass m is the tension in the cord. Therefore, the magnitude of the centripetal force is equal to the tension in the cord.
On the other hand, the centrifugal force acts in the opposite direction of the tension in the cord and has a magnitude equal to the mass of the cylinder (M) times the acceleration due to gravity (g).
To keep the cylinder at rest, the net force acting on it must be zero. This means that the magnitude of the tension in the cord must be equal to the magnitude of the centrifugal force.
Mathematically, we can express this as:
T = M * g
Now, let's relate the tension to the speed of mass m. The centripetal force is given by:
F = m * v^2 / r
where v is the speed of mass m and r is the radius of the circle.
Substituting T for F, we have:
T = m * v^2 / r
Since we want the speed at which the tension is equal to the weight of the cylinder, we can substitute M * g for T:
M * g = m * v^2 / r
Now, we can solve for v:
v^2 = (M * g * r) / m
v = sqrt[(M * g * r) / m]
Substituting the given values:
v = sqrt[(2.5 kg * 9.8 m/s^2 * 0.20 m) / 1.5 kg]
v = sqrt[32.67 m^2/s^2]
v ≈ 5.72 m/s
Therefore, the speed of mass m that keeps the cylinder at rest is approximately 5.72 m/s.
To keep the cylinder in equilibrium both side must be equal.
(m1)*(v^2/r)=(m2)*g Left side=Right side
(v^2/r) = (m2)*g / m1 isolate “v” on the left…
v^2 = (m12)*g*r/m1
v = sqrt((m2)*g*r /(m1)) plug in the values and you are done…
v = sqrt(2.5*9.81*.2)/(1.5) = 1.80831
Dr. Cooper will be proud of you.
m1 = mass of the puck = 1.5 km
m2 = mass of the cylinder = 2.5 km
r= radius = 20 cm = .2 m (SI units)
v = velocity
g = 9.81 (if do not know what “g” is you should look it up)
sqrt = square root