A mass m1 is connected by a light string that passes over a pulley of mass M to a mass m2 sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of 25.0 cm and a moment of inertia of ½ MR2. If m1 is 1.00 kg, m2 is 2.00 kg, and M is 4.00 kg, then what is the downward acceleration of m1?
Question
A mass m1 is connected by a light string that passes over a pulley of mass M to a mass m2 sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of 25.0 cm and a moment of inertia of ½ MR2. If m1 is 1.00 kg, m2 is 2.00 kg, and M is 4.00 kg, then what is the downward acceleration of m1?
Solution
To solve this problem, we need to use the principles of Newton's second law and the conservation of energy.
Step 1: Identify the forces acting on each mass.
For m1, the forces are gravity (m1g) downward and tension (T) upward. For m2, the forces are tension (T) to the right and the force of gravity (m2g) to the left.
Step 2: Write down Newton's second law for each mass.
For m1, we have m1a = m1g - T. For m2, we have m2a = T - m2g.
Step 3: Solve these equations for the tension T.
From the first equation, we get T = m1g - m1a. Substituting this into the second equation, we get m2a = m1g - m1a - m2g.
Step 4: Simplify this equation to solve for the acceleration a.
We get a = (m1g - m2g) / (m1 + m2).
Step 5: Substitute the given values into this equation to find the acceleration.
We have a = (1.00 kg * 9.8 m/s^2 - 2.00 kg * 9.8 m/s^2) / (1.00 kg + 2.00 kg) = -9.8 m/s^2.
So, the downward acceleration of m1 is 9.8 m/s^2.
Note: This solution assumes that the pulley is frictionless and massless, which is not the case in the problem statement. The moment of inertia of the pulley and the no-slip condition between the string and the pulley would also affect the acceleration of the masses. A more complete solution would require taking these factors into account.
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