In the figure provided, a 295 kg crate, to the right of the pulley, is released from rest and begins to fall to the ground. If the mass on the left is m = 215 kg, what is the magnitude of the acceleration of the 295 kg crate? Assume the rope and pulley are massless.
Question
In the figure provided, a 295 kg crate, to the right of the pulley, is released from rest and begins to fall to the ground. If the mass on the left is m = 215 kg, what is the magnitude of the acceleration of the 295 kg crate? Assume the rope and pulley are massless.
Solution
To solve this problem, we need to use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Step 1: Identify the forces acting on each mass.
For the 295 kg crate, there are two forces acting on it: the force of gravity pulling it downwards (F1 = m1*g = 295 kg * 9.8 m/s^2 = 2891 N) and the tension in the rope pulling it upwards (T).
For the 215 kg mass, there are also two forces acting on it: the force of gravity pulling it downwards (F2 = m2*g = 215 kg * 9.8 m/s^2 = 2107 N) and the tension in the rope pulling it upwards (T).
Step 2: Set up the equations of motion for each mass.
For the 295 kg crate, the equation of motion is F1 - T = m1*a.
For the 215 kg mass, the equation of motion is T - F2 = m2*a.
Step 3: Solve the equations of motion for the acceleration (a).
Adding the two equations of motion gives F1 - F2 = (m1 + m2)*a.
Solving for a gives a = (F1 - F2) / (m1 + m2) = (2891 N - 2107 N) / (295 kg + 215 kg) = 784 N / 510 kg = 1.54 m/s^2.
So, the magnitude of the acceleration of the 295 kg crate is 1.54 m/s^2.
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