two masses m' each are hanging with help of a light string,which passes over mass less pulley. mass A starts leaking out at a rate of u kg/s with velocity v' with respect to mass A. find kinetic energy of block B as a function of time.
To find the kinetic energy of block B as a function of time, we need to consider the forces acting on both blocks and use Newton's laws of motion.
Let's break down the problem step by step:
1. Define the variables:
- m' = mass of each block (m' > 0)
- u = leaking rate of mass A (u > 0, kg/s)
- v' = velocity of leaked mass A with respect to mass A (v' > 0, m/s)
2. Determine the forces:
- Gravitational force acting on mass A is m' * g, where g is the acceleration due to gravity (9.8 m/s²).
- Tension in the string is the same on both ends, so block B also experiences a force of m' * g.
3. Apply Newton's second law:
- For mass A: m' * g - u * v' = m' * a₁, where a₁ is the acceleration of mass A.
- For mass B: m' * g = m' * a₂, where a₂ is the acceleration of mass B. Block B doesn't experience any external force other than gravity.
4. Relate the accelerations:
- Since both ends of the string move with the same velocity (v'), the accelerations of mass A and block B are the same. Therefore, a₁ = a₂ = a.
5. Solve the differential equation:
- Rearrange the equations:
- m' * (g - u * v') = m' * a
- m' * g = m' * a
- Simplify by dividing both equations by m':
- g - u * v' = a
- g = a
- Since a = g, acceleration is constant and equal to g.
- Integrate to get the velocity as a function of time:
- v = g * t + C, where v is the velocity of block B, t is time, and C is the constant of integration.
6. Determine the constant C:
- At t = 0, the initial velocity of block B is 0 because it is at rest.
- Substitute v = 0 and t = 0 into the equation:
- 0 = g * 0 + C
- C = 0
7. Final velocity as a function of time:
- Substitute C = 0 into the equation:
- v = g * t
8. Kinetic energy of block B:
- The kinetic energy (KE) of an object is given by KE = (1/2) * m * v².
- Substitute m = m' and v = g * t into the equation:
- KE = (1/2) * m' * (g * t)²
- KE = (1/2) * m' * g² * t²
Therefore, the kinetic energy of block B as a function of time is given by KE = (1/2) * m' * g² * t².