Answer to Question #105195 in Classical Mechanics for Andsson

Question

4. A student chased by the bull runs through the gate with uniform mass of 153 Kg. The gate is 2.8 m wide and 1.5 m high. The bull will escape if the gate is not closed in 30 sec. The student pushes the gate with 100N force directed perpendicular to the line at its edge 2.8 m from the hinge. a. Calculate the moment of inertia of the gate. b. Angular acceleration of the gate c. Time needed to close the gate if it starts at rest and rotates through 50 before closing.

5. A 15-kg object and a 10-kg object are suspended, joined by a cord that passes over a pulley with a radius of 15 cm and a mass of 3 kg. The cord has a negligible mass and does not slip on the pulley. Treat the pulley as a uniform disk, and determine the linear acceleration of the two objects after the objects are released from rest.

Answer 1

4. a. The moment of inertia of the gate is

$$I=\frac{1}{3}ml^2=\frac{1}{3}\cdot153\cdot2.8^2=399.84\text{ kg}\cdot\text{m}^2.$$

b. The angular acceleration is

$$\alpha=\frac{\tau}{I}=\frac{Fl}{I}=\frac{100\cdot2.8}{399.84}=0.7\text{ rad/s}^{2}.$$

c. For a 50-degree angle, the time will be

$$t=\sqrt{\frac{2\phi}{\alpha}}=\sqrt{\frac{2\cdot50^\circ/180^\circ\cdot3.14}{0.7}}=1.58\text{ s}.$$

5. The moment of inertia of the pulley:

$$I=\frac{1}{2}mr^2.$$

Newton's second law, torque equilibrium and angular to linear acceleration transformation for the two blocks and the disk:

$$T_1-m_1g=-m_1a,\\ T_2-m_2g=m_2a,\\ T_1R-T_2R=I\alpha,\\ \space\\ \alpha=\frac{a}{r}.$$

Solving this altogether, we have

$$a=g\frac{m_1-m_2}{m_1+m_2+\frac{1}{2}m}=1.85\text{ m/s}^2.$$