A puck of mass m = 1.10 kg slides in a circle of radius r = 16.9 cm on a frictionless table while attached to a hanging cylinder of mass M = 2.20 kg by means of a cord that extends through a hole in the table. What speed keeps the cylinder at rest?
Ac = v^2/r
1.1 Ac = 2.2 g
so
Ac = (2.2/1.1)(9.81) = 19.6 m/s^2
v^2 = .169 (19.6) = 3.32
v = 1.82 meters/second
that is the speed
if you want angular velocity
omega = v/r = 1.82 /.169 = 10.8 radians/second
thanks so much!!!!!
To determine the speed that keeps the cylinder at rest, we need to consider the forces acting on the system.
1. First, let's identify the forces acting on the puck:
- Tension force: This force is exerted by the cord and provides the centripetal force required to keep the puck moving in a circle.
- Weight force: The puck experiences a gravitational force due to its mass, given by the equation F_gravity = m * g, where m is the mass of the puck and g is the acceleration due to gravity.
2. Now, let's consider the forces acting on the cylinder:
- Tension force: The cord exerts an upward tension force on the cylinder.
- Weight force: The cylinder experiences a gravitational force due to its mass, given by the equation F_gravity = M * g, where M is the mass of the cylinder and g is the acceleration due to gravity.
- Reaction force: The table exerts an equal and opposite reaction force on the cylinder to support its weight.
Since the system is frictionless, there is no friction force to consider.
3. The key idea is that the tension force throughout the cord remains constant, as it is the same cord. Therefore, the tension force acting on both the puck and the cylinder must be equal.
Using Newton's second law, we can equate the centripetal force required for the puck with the tension force in the cord:
Tension force = Centripetal force
m * v^2 / r = Tension force
Where:
- m is the mass of the puck,
- v is the speed of the puck,
- r is the radius of the circle.
Since the tension force is the same throughout the cord, it will also be the force acting upward on the cylinder:
Tension force = M * g
Setting these two equations equal to each other and solving for v:
m * v^2 / r = M * g
Rearranging the equation:
v^2 = (M * g * r) / m
Taking the square root of both sides:
v = sqrt((M * g * r) / m)
Now we can substitute the given values:
- m = 1.10 kg (mass of the puck)
- M = 2.20 kg (mass of the cylinder)
- r = 16.9 cm = 0.169 m (radius of the circle)
- g = 9.8 m/s^2 (acceleration due to gravity)
v = sqrt((2.20 kg * 9.8 m/s^2 * 0.169 m) / 1.10 kg)
Calculating the value of v will give us the speed that keeps the cylinder at rest.