Ok so this question was posted before but I'm confused by the answer. The problem is: The two masses (m1 = 5.0 kg and m2 = 3.0 kg) in the Atwood's machine shown in Figure 10-23 are released from rest, with m1 at a height of 0.87 m above the floor. When m1 hits the ground its speed is 1.4 m/s. Assume that the pulley is a uniform disk with a radius of 12 cm. Find the mass of the pulley.
Since the pulley is a disk that means I = 1/2mr^2 and omega is vfinal/r. I have my equation 1/2(m2 + m1)vfinal^2 + 1/2Iw^2 = m1gh but I don't know what I'm doing wrong.
Solve for I in that equation, then from
knowing the value of I, set it equal to 1/2 masspulley*radius^2 and solve for the masso of the pulley. radius pulley=.12m
To solve this problem, you can use the law of conservation of mechanical energy. Let's break down the steps to find the mass of the pulley:
Step 1: Identify the known values:
- Mass of m1: m1 = 5.0 kg
- Mass of m2: m2 = 3.0 kg
- Height of m1 above the ground: h = 0.87 m
- Final velocity of m1: vfinal = 1.4 m/s
- Radius of the pulley: r = 12 cm = 0.12 m
Step 2: Calculate the gravitational potential energy (GPE) of m1 at its initial position:
GPE = m1 * g * h
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 3: Calculate the moment of inertia (I) of the pulley:
Since the pulley is a uniform disk, the moment of inertia (I) can be calculated as I = (1/2) * m_pulley * r^2. We want to find the mass of the pulley (m_pulley) in this step.
Step 4: Calculate the final angular velocity (ω) of the pulley:
The final angular velocity (ω) can be calculated as ω = vfinal / r,
where vfinal is the final velocity of m1.
Step 5: Calculate the kinetic energy (KE) of m1 just before it hits the ground:
KE = (1/2) * m1 * vfinal^2
Step 6: Apply the law of conservation of mechanical energy (GPE + KE = 1/2 m_total v^2):
GPE + KE + (1/2) I ω^2 = 0
Since the pulley is balanced, the tensions in the two strings cancel each other out and do not affect the energy equation.
Step 7: Substitute the known values into the equation:
(m1 * g * h) + (1/2) * m1 * vfinal^2 + (1/2) * [(1/2) * m_pulley * r^2] * (vfinal / r)^2 = 0
Step 8: Solve for m_pulley, the mass of the pulley:
Rearrange the equation and solve for m_pulley. You will end up with an equation in terms of m_pulley. Simplify and solve for m_pulley.
By following these steps, you will be able to find the mass of the pulley in the given Atwood's machine problem.