A puck of mass m = 1.26 kg slides in a circle of radius r = 25.0 cm on a frictionless table while attached to a hanging cylinder of mass M = 2.04 kg by a cord through a hole in the table. What speed keeps the cylinder at rest?
_________m/s
Mg=mv^2/r
solve for v.
To find the speed required to keep the cylinder at rest, we need to consider the forces acting on the system.
The only force acting on the system is the tension in the cord. At the instant when the cylinder is at rest, the tension in the cord balances the gravitational force acting on it.
The tension in the cord can be calculated using the centripetal force equation:
Tension = (Mass of puck) * (Centripetal acceleration)
The centripetal acceleration can be calculated using the equation:
Centripetal acceleration = (Velocity^2) / (Radius)
Since the puck is sliding in a circle of radius 25.0 cm, the centripetal acceleration is given by:
Centripetal acceleration = (Velocity^2) / (0.25 m)
Finally, we can write the equation for the tension in the cord as:
Tension = (Mass of puck) * [(Velocity^2) / (0.25 m)]
However, since we want to find the speed that keeps the cylinder at rest, the tension in the cord must be equal to the gravitational force acting on the cylinder:
Tension = (Mass of cylinder) * (Gravity)
Combining these equations, we have:
(Mass of puck) * [(Velocity^2) / (0.25 m)] = (Mass of cylinder) * (Gravity)
Now, we can substitute the given values into the equation and solve for the velocity:
(1.26 kg) * [(Velocity^2) / (0.25 m)] = (2.04 kg) * (9.8 m/s^2)
Simplifying the equation:
[(Velocity^2) / (0.25 m)] = (2.04 kg * 9.8 m/s^2) / (1.26 kg)
[(Velocity^2) / (0.25 m)] = 15.86 m/s^2
Velocity^2 = 15.86 m/s^2 * 0.25 m
Velocity^2 = 3.965 m^2/s^2
Taking the square root of both sides:
Velocity ≈ √(3.965 m^2/s^2) ≈ 1.99 m/s
Therefore, the speed required to keep the cylinder at rest is approximately 1.99 m/s.
To find the speed that keeps the cylinder at rest, we need to consider the forces acting on the system.
In this setup, the only two forces acting on the system are the tension in the cord and the weight of the cylinder. Since the cord is attached horizontally, the tension in the cord provides the centripetal force required for circular motion.
The tension in the cord can be determined by equating it to the force required to keep the puck moving in a circular path with radius r. The centripetal force is given by the equation:
F = m * v^2 / r
Where,
m = mass of the puck
v = speed of the puck
r = radius of the circular path
In this case, the centripetal force is provided by the tension in the cord. So we can rewrite the equation as:
Tension = m * v^2 / r ----(1)
Now, let's consider the other force acting on the system, which is the weight of the cylinder. The weight is given by the equation:
Weight = mass * gravity
Where,
mass = mass of the cylinder
gravity = acceleration due to gravity
The weight of the cylinder acts in the downward direction. Since the cylinder is at rest, the tension in the cord must balance the weight of the cylinder. Therefore, we can write:
Tension = Weight of the cylinder ----(2)
Substituting the values into equations (1) and (2), we get:
m * v^2 / r = M * gravity
Rearranging the equation to solve for v, we get:
v^2 = (M * gravity * r) / m
Taking the square root of both sides, we finally get the formula to calculate the speed:
v = sqrt((M * gravity * r) / m)
Now we can substitute the given values into this equation to find the speed that keeps the cylinder at rest:
m = 1.26 kg
r = 0.25 m (converted from 25.0 cm)
M = 2.04 kg
gravity = 9.8 m/s^2
Plugging in the values, we get:
v = sqrt((2.04 kg * 9.8 m/s^2 * 0.25 m) / 1.26 kg)
Simplifying this equation, we find:
v ≈ 1.69 m/s
Therefore, the speed that keeps the cylinder at rest is approximately 1.69 m/s.