Red ribbon wound around a red spool (above) is taped to blue ribbon wound around a blue spool (below). The ribbon is essentially massless, but the solid cylindrical spools each have mass m and radius R. If the red spool can freely rotate on a fixed axle and the blue spool is positioned directly underneath with the ribbon taut, what is the downward acceleration of the blue spool? Gravity is downward.
Enter your answer in terms of some or all of the variables m, R and g.
To find the downward acceleration of the blue spool, we can consider the forces acting on it.
1. Tension in the ribbon: The tension in the ribbon acts vertically downward due to the weight of the red spool. We can assume that the tension in the ribbon is constant throughout its length.
2. Weight of the red spool: The weight of the red spool acts vertically downward.
Since the blue spool is taut to the red spool, the downward acceleration of the blue spool will be the same as the red spool. Let's calculate the downward acceleration of the red spool.
The net force on the red spool is given by the difference between the weight of the red spool and the tension in the ribbon.
Net force on the red spool = Weight of the red spool - Tension in the ribbon
The weight of the red spool is given by W_s = m * g, where m is the mass of the red spool and g is the acceleration due to gravity.
The tension in the ribbon is equal to the weight of the red spool since the ribbon is massless.
Therefore, the net force on the red spool is:
Net force on the red spool = m * g - m * g = 0
Since the net force on the red spool is zero, there is no acceleration or deceleration in the vertical direction. Therefore, the downward acceleration of the blue spool is also zero.
To calculate the downward acceleration of the blue spool, we need to consider the forces acting on it. In this case, we have two main forces: the tension force in the ribbon and the gravitational force.
Let's start by analyzing the tension force in the ribbon. Since the ribbon is taut, the tension force in the ribbon will be the same throughout its entire length. This tension force is what causes the acceleration of the blue spool.
Now, think about the red spool. As it rotates, the ribbon wound around it also rotates. This means that the tension force in the ribbon leads to a torque on the red spool, causing it to rotate. According to Newton's third law, the red spool exerts an equal and opposite torque on the blue spool.
The torque exerted by the red spool on the blue spool is given by the equation:
Torque = Moment of Inertia x Angular Acceleration
Since the red spool is freely rotating, it has a moment of inertia equal to ½ mR² (for a solid cylinder). The angular acceleration of the red spool is related to the downward acceleration of the blue spool:
Angular Acceleration = Acceleration / R
Now, let's analyze the gravitational force. The gravitational force acting on the blue spool is equal to the mass of the blue spool (m) times the acceleration due to gravity (g). This force acts straight downward.
Since the torque exerted by the red spool is equal to the gravitational torque, we can equate the two:
Torque = Gravitational Torque
(½ mR²) x (Acceleration / R) = m x g x R
Simplifying this equation, we find:
Acceleration = (2g)
Therefore, the downward acceleration of the blue spool is equal to 2g, where g represents the acceleration due to gravity.