Two objects (53.0 and 16.0 kg) are connected by a massless string that passes over a massless, frictionless pulley. The pulley hangs from the ceiling. Find (a) the acceleration of the objects and (b) the tension in the string.
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To find the acceleration of the objects and the tension in the string, we can use Newton's second law of motion.
(a) The acceleration can be found by comparing the gravitational force on the two objects with the net force acting on the system.
Let's assume that the 53.0 kg object is hanging downward on one side of the pulley, and the 16.0 kg object is being pulled upward by the string on the other side of the pulley.
The gravitational force on the 53.0 kg object can be calculated using the equation F = m * g, where m is the mass and g is the acceleration due to gravity (usually 9.8 m/s^2). So, for the 53.0 kg object, the force acting downwards is F1 = (53.0 kg) * (9.8 m/s^2).
On the other side of the pulley, the 16.0 kg object is being pulled upwards by the tension in the string. So, the force acting upwards is F2 = Tension.
According to Newton's second law, the net force acting on the system is equal to the product of the total mass and the acceleration of the system. Since the two objects are connected by the string, they have the same acceleration. So, the net force acting on the system is given by F_net = (53.0 kg + 16.0 kg) * acceleration.
Since the objects are connected by a string passing over a pulley, the tension in the string is the same throughout. It can be represented by the variable "Tension".
To set up the equation, we'll assume that the 53.0 kg object moves downward, while the 16.0 kg object moves upward. Therefore, the total force acting on the system is the difference between the two forces: F_net = F1 - F2.
Setting up the equation and solving for acceleration:
(53.0 kg + 16.0 kg) * acceleration = F1 - F2
(69.0 kg) * acceleration = (53.0 kg) * (9.8 m/s^2) - (Tension)
(b) To find the tension in the string, we need to solve for it in the equation we obtained in part (a).
To do so, rearrange the equation to solve for tension:
Tension = (53.0 kg) * (9.8 m/s^2) - (69.0 kg) * acceleration.
Now you can substitute the value of acceleration into this equation (from part (a)) and calculate the tension.