Energy Conservation Method

Q. A solid uniform cylinder of mass $M$ performs small oscillations in the horizontal plane if slightly displaced from its mean position as shown in the figure. Initially, the springs are at their natural lengths and the cylinder does not slip on the ground during oscillations due to friction between the ground and cylinder. The force constant of each spring is $k$. What is the time period of this oscillation?

1. $\pi \sqrt{\frac{3M}{k}}$
2. $\pi \sqrt{\frac{3M}{4k}}$
3. $\frac{\pi }{2}\sqrt{\frac{3M}{k}}$
4. $\frac{3\pi }{2}\sqrt{\frac{M}{k}}$

Q. One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to a massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross – section and Young’s modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period given by

1. $2\pi \sqrt{\frac{m}{k}}$
2. $2\pi {\left[\frac{(YA+KL)m}{YAK}\right]}^{\frac{1}{2}}$
3. $2\pi \left(\frac{mYA}{KL}\right)$
4. $2\pi \left(\frac{mL}{YA}\right)$

Q. A solid uniform cylinder of mass $M$ performs small oscillations in the horizontal plane if slightly displaced from its mean position as shown in the figure. Initially, the springs are at their natural lengths and the cylinder does not slip on the ground during oscillations due to friction between the ground and cylinder. The force constant of each spring is $k$. What is the time period of this oscillation?

1. $\pi \sqrt{\frac{3M}{k}}$
2. $\pi \sqrt{\frac{3M}{4k}}$
3. $\frac{\pi }{2}\sqrt{\frac{3M}{k}}$
4. $\frac{3\pi }{2}\sqrt{\frac{M}{k}}$