An object (m=2.85 kg) is attached to a disk-shaped pulley. The diameter of the pulley is 0.242 and its weight is 7.272 N. Suppose the object is allowed to fall (a) What is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 2.0 s?
To find the answers to these questions, we will use some basic concepts from physics such as Newton's second law, torque, and kinematic equations.
(a) To find the linear acceleration of the object, we can use Newton's second law, which states that the net force on an object is equal to the mass of the object times its acceleration (F = m * a). In this case, the only force acting on the object is its weight, so we can write down the equation:
F = m * a
Where F is the force due to the weight of the object, which is equal to its mass times the acceleration due to gravity (F = m * g). Taking the magnitude of the force due to gravity as 9.8 m/s^2, we can substitute the values into the equation:
m * a = m * g
2.85 kg * a = 2.85 kg * 9.8 m/s^2
Solving for the acceleration 'a', we find:
a = 9.8 m/s^2
Therefore, the linear acceleration of the object is 9.8 m/s^2.
(b) To find the angular acceleration of the pulley, we need to consider the torque exerted on the pulley. Torque is calculated by multiplying the force applied to an object by the perpendicular distance from the axis of rotation to the line of action of the force. In this case, the torque is due to the weight of the pulley, so we can write down the equation:
Torque = Force * Radius
The force acting on the pulley is its weight, which is given as 7.272 N. The radius of the pulley is half of its diameter, so we can calculate it as:
Radius = Diameter / 2 = 0.242 m / 2 = 0.121 m
Substituting the values into the equation, we find:
Torque = 7.272 N * 0.121 m
To calculate angular acceleration, we need to use the formula:
Torque = Moment of Inertia * Angular Acceleration
The moment of inertia of a disk is given by:
Moment of Inertia = (1/2) * mass * radius^2
Substituting the values into the equation, we find:
7.272 N * 0.121 m = (1/2) * m * r^2 * angular acceleration
0.881112 N*m = (1/2) * (2.85 kg) * (0.121 m)^2 * angular acceleration
Simplifying further, we can solve for the angular acceleration 'α':
angular acceleration = (2 * 0.881112 N*m) / ((2.85 kg) * (0.121 m)^2)
Therefore, the angular acceleration of the pulley is approximately 6.63 rad/s^2.
(c) To find how far the bucket drops in 2.0 seconds, we can use the kinematic equation that relates displacement, initial velocity, time, and acceleration:
displacement = (1/2) * acceleration * time^2
Substituting the values into the equation, we find:
displacement = (1/2) * 9.8 m/s^2 * (2.0 s)^2
Therefore, the bucket drops approximately 19.6 meters in 2.0 seconds.