A simple pendulum is a basic mechanical system consisting of a mass (bob) suspended by a string or rod of negligible mass. When the mass is displaced from its equilibrium position, it moves in a circular arc under the influence of gravity.

The pendulum oscillates in periodic motion around the equilibrium point, and the study of the simple pendulum is fundamental to understanding simple harmonic motion.

The correct answer is: B) A mechanical system consisting of a mass suspended by a massless string

The period of a simple pendulum depends only on the length of the string and the gravitational acceleration according to the relation:

\( T = 2\pi \sqrt{\frac{L}{g}} \)

where T is the period, L is the length of the string, and g is the gravitational acceleration.

Galileo discovered that the period of a pendulum does not depend on the mass of the suspended bob, which is one of the fundamental discoveries in pendulum physics.

The correct answer is: B) Length of the string

Kinetic energy is at its maximum when the pendulum's speed is at its maximum, which occurs at the lowest point of the path (equilibrium position).

At the lowest point, potential energy is at its minimum and kinetic energy is at its maximum according to the principle of energy conservation.

While at the highest point, kinetic energy is zero and potential energy is at its maximum.

The correct answer is: B) At the lowest point (equilibrium position)

The correct relationship for the period of a simple pendulum is:

\( T = 2\pi \sqrt{\frac{L}{g}} \)

where T is the period, L is the length of the string, and g is the gravitational acceleration.

This relationship applies only for small angles (less than 15 degrees). For large angles, the motion becomes more complex and this simple relationship does not apply.

The correct answer is: B) \( T = 2\pi \sqrt{\frac{L}{g}} \)

The height of the pendulum from the equilibrium position is calculated using the relation:

\[ h = L - L \cos \theta = L (1 - \cos \theta) \]

where h is the height, L is the length of the string, and θ is the displacement angle.

This relationship is derived from calculating the difference between the string length and the vertical distance of the pendulum from the suspension point.

The correct answer is: A) \[ h = L - L \cos \theta \]

From the relation \[ T = 2\pi \sqrt{\frac{L}{g}} \], we notice that the period is proportional to the square root of the length.

If the length increases to 4 times, the period will increase to \[ \sqrt{4} = 2 \] times.

Therefore, if the original period is T, the new period will be 2T, and the ratio is 2:1.

The correct answer is: B) Becomes twice

To calculate the maximum velocity, we use the principle of energy conservation:

Potential energy at maximum height = Kinetic energy at the lowest point

\[ m g h = \frac{1}{2} m v^2 \]

We calculate the height \[ h = L (1 - \cos \theta) = 1 \times (1 - \cos 30^\circ) = 1 \times (1 - 0.866) = 0.134 m\]

Substituting: \[ 9.8 \times 0.134 = \frac{1}{2} v^2 \]

\[ 1.3132 = \frac{1}{2} v^2 \rightarrow v^2 = 2.6264 \rightarrow v = \sqrt{2.6264} = 1.62 m/s\]

The correct answer is: B) 1.63 m/s (rounded)