The system shown in the figure below consists of a m1 = 5.52-kg block resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging m2 = 2.76-kg block.The pulley is a uniform disk of radius 7.86 cm and mass 0.592 kg. Calculate the speed of the m2 = 2.76-kg block after it is released from rest and falls a distance of 2.08 m. Calculate the angular speed of the pulley at this instant.
To find the speed of the m2 block after it falls a distance of 2.08 m, we can use the concept of conservation of energy.
1. Calculate the potential energy gained by the m2 block:
- Potential energy gained = mass of m2 block * acceleration due to gravity * height
- Potential energy gained = 2.76 kg * 9.81 m/s^2 * 2.08 m
2. Calculate the potential energy lost by the m1 block:
- Potential energy lost = mass of m1 block * acceleration due to gravity * height
- Potential energy lost = 5.52 kg * 9.81 m/s^2 * 2.08 m
3. The potential energy lost by the m1 block is transferred to the kinetic energy gained by the m2 block:
- Potential energy lost = Kinetic energy gained by m2 block
- 5.52 kg * 9.81 m/s^2 * 2.08 m = 2.76 kg * v^2/2
- Solve for v (the speed of the m2 block)
4. Once you find the speed of the m2 block, you can use it to calculate the angular speed of the pulley.
- The linear velocity of the outer edge of the pulley is equal to the radius times the angular velocity.
- Linear velocity = radius * angular velocity
- v = 0.0786 m * ω (where ω is the angular velocity)
- Solve for ω (the angular speed of the pulley)
Now, you can calculate the speed of the m2 block and the angular speed of the pulley using the above equations and the given values for mass, radius, and height.