Two blocks are connected with a light string that is hung over a pulley, as shown in the diagram to the right. One block has twice the mass of the other, and the pulley
has the same mass as the heavier block. The pulley consists of a solid, uniform circular disk. The heavier block is positioned a distance of 2.5m higher than the
lighter block, and they are released from rest. Assume the string does not slip across the surface of the pulley. Use mechanical energy conservation to find the speed of the blocks at the point when they swap positions (i.e. the heavier block is 2.5m lower than the lighter block)
To find the speed of the blocks when they swap positions, we can use the principle of mechanical energy conservation. Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE). At the point when the heavier block is 2.5m lower than the lighter block, all of their initial potential energy will be converted into kinetic energy.
Here's how to approach it step by step:
1. Determine the initial potential energy (PE_initial) of the system:
- The potential energy of the heavier block is given by: PE_h = m_h * g * h_h
Here, m_h represents the mass of the heavier block and h_h is the height difference between the initial positions of the blocks.
- The potential energy of the lighter block is given by: PE_l = m_l * g * h_l
Here, m_l represents the mass of the lighter block and h_l is the initial height of the lighter block.
- The total initial potential energy of the system is: PE_initial = PE_h + PE_l = (m_h * g * h_h) + (m_l * g * h_l)
2. Determine the final kinetic energy (KE_final) of the system:
- When the blocks swap positions, they will have the same speed. Let's call this speed v.
- The final kinetic energy of the heavier block is given by: KE_h = (1/2) * m_h * v^2
- The final kinetic energy of the lighter block is given by: KE_l = (1/2) * m_l * v^2
- The total final kinetic energy of the system is: KE_final = KE_h + KE_l = (1/2) * (m_h + m_l) * v^2
3. Apply the principle of mechanical energy conservation:
- According to the principle, the initial potential energy of the system is equal to the final kinetic energy of the system:
PE_initial = KE_final
- Substitute the expressions for PE_initial and KE_final calculated in steps 1 and 2, respectively.
- Solve the equation for v^2.
4. Calculate the speed (v):
- Take the square root of v^2 to find the speed v.
Note that in this problem, the mass of the pulley is given to be the same as the heavier block. If the mass of the pulley were different, additional considerations would be necessary.