Two blocks of masses m = 4 kg and M = 6 kg are connected by a light rope that is swung over a smooth pulley hanging from the roof. At the initial moment block M is at a height of H = 1 meter from the floor. Both blocks are released from rest, but block m is initially at an unknown height from the floor. With what speed will block M strike the floor?
Question
Two blocks of masses m = 4 kg and M = 6 kg are connected by a light rope that is swung over a smooth pulley hanging from the roof. At the initial moment block M is at a height of H = 1 meter from the floor. Both blocks are released from rest, but block m is initially at an unknown height from the floor. With what speed will block M strike the floor?
Solution
To solve this problem, we can use the principle of conservation of mechanical energy.
Step 1: Determine the potential energy of block M at the initial moment. The potential energy is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the mass of block M is 6 kg and the height is 1 meter. So, the potential energy of block M is PE = 6 kg * 9.8 m/s^2 * 1 m = 58.8 J.
Step 2: Determine the potential energy of block m at the initial moment. Since the height of block m is unknown, we can represent it as h_m. So, the potential energy of block m is PE = mgh_m.
Step 3: Determine the kinetic energy of block M just before it strikes the floor. At this point, the potential energy of block M is zero because it has reached the floor. Therefore, all of its initial potential energy is converted into kinetic energy. The kinetic energy is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity. In this case, the mass of block M is 6 kg. So, the kinetic energy of block M is KE = 58.8 J.
Step 4: Equate the initial potential energy of block M to its final kinetic energy. This gives us the equation mgh = (1/2)mv^2. We can cancel out the mass m on both sides of the equation.
Step 5: Solve for the velocity v. Rearranging the equation, we get v^2 = 2gh. Taking the square root of both sides, we get v = sqrt(2gh).
Step 6: Substitute the known values into the equation. The acceleration due to gravity is approximately 9.8 m/s^2 and the height h is 1 meter. So, v = sqrt(2 * 9.8 m/s^2 * 1 m) = sqrt(19.6) m/s.
Step 7: Calculate the final answer. Using a calculator, we find that the speed at which block M strikes the floor is approximately 4.43 m/s.
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