Question

A 0.20-kg block and a 0.25-kg block are connected to each other by a string draped over a pulley that is a solid disk of inertia 0.50 kg and radius 0.10 m. When released, the 0.25-kg block is 0.21 m off the ground. What speed does this block have when it hits the ground?

Answer 1

Answer:

The answer is "0.2711  m/s".

Explanation:

Potential energy = Kinetic energy + Potential energy

$m_1 gh =\frac{1}{2} m_1v^2 +\frac{1}{2} m_2v^2 + \frac{1}{2} I\omega^2 + m_1gh$

$(m_1- m_2)gh =\frac{1}{2} m_1v^2 +\frac{1}{2} m_2v^2 +\frac{1}{2} I\omega^2\\\\2(m_1 - m_2)gh = m_1v^2 + m_1v^2 + I\omega^2\\\\solid \ disk (I) = \frac{1}{2} \ \ M r^2$

When there is no slipping, \omega  =\frac{ v]{r}\\\\  

$2(m_1 - m_2)gh = m_1v^2 + m_2v^2 + (\frac{1}{2} Mr^2) (\frac{v}{r})^2\\\\2(m_1 -m_2)gh = m_1v^2 + m_2v^2 + \frac{1}{2} Mv^2\\\\4(m_1 -m_2)gh = 2m_1v^2 + 2m_2v^2 + Mv^2\\\\4(m_1 - m_2)gh = (2m_1 + 2m_2 + M) v^2$

$v^2 = \frac{4(m_1 - m_2)gh}{(2m_1 + 2m_2 + M)}v$

$v^2 = \frac{4 (0.25 \ kg - 0.20 \ kg) (9.8 \frac{m}{s^2}) (0.21 m)}{ (2 \times 0.25 kg + 2 \times 0.20 kg + 0.50 kg)}$

$=\frac{0.1029}{1.4} \ \ \frac{m^2}{s^2}\\\\=0.0735\ \ \frac{m^2}{s^2}\\\\= 0.2711 \ \frac{m}{s}$