Assertion (A) is true: On an unbanked road, the maximum safe velocity is \( v_{\text{max}} = \sqrt{mu_s gr} \). An increase in friction (\( \mu_s \)) leads to an increased \( v_{\text{max}} \).

Reason (R) is true: Banking of roads provides a component of the normal force for centripetal force, effectively increasing the limiting velocity. (R) is not the correct explanation for (A) as they represent different factors influencing safe velocity.

Assertion (A) is true: As the angular speed \( \omega \) of the gramophone increases, the required centripetal force \( m \omega^2 r \) for the coin increases. When this force exceeds the maximum static friction (\( \mu_s mg \)), the coin slips off.

Reason (R) is false: The centripetal force is provided by the frictional force between the coin and the gramophone, not by the gravitational force. Gravitational force provides the normal force.

The absolute speed of the trains relative to Earth's center is \( v_{\text{abs}} = R_E \omega_E pm v_{\text{surface}} \). Since their absolute speeds are different, the centripetal force required \( F_c = m v_{\text{abs}}^2 / R_E \) will be different. The normal reaction is \( N = mg - F_c \), so it will also be different. Therefore, both Assertion (A) and Reason (R) are false.

In non-uniform circular motion, there is a tangential component of force, which causes a change in speed. Work done is `\(W = \int \vec{F}_{net} \cdot d\vec{r}\)`.
Since the net force is not always perpendicular to the elementary displacement `\(d\vec{r}\)` due to the tangential component, the work done by the net force is not zero. Both assertion and reason are true, and the reason correctly explains the assertion.

Angular velocity \(\omega = \frac{2\pi}{\text{T}}\). For a seconds hand, \(\text{T} = 60\text{ s}\). Thus, \(\omega = \frac{2\pi}{60} = \frac{\pi}{30}\text{ rad/s}\). Both assertion and reason are true, and the reason correctly explains the assertion.

In uniform circular motion, while the magnitudes of velocity and centripetal acceleration are constant, their directions continuously change. Thus, both vectors \(\vec{v}\) and \(\vec{a}\) are non-constant.

For an ideal frictionless banked curve, the optimum speed is given by \(v = \sqrt{gRtan\theta}\). Given R = 90 m, \(\theta = 45^\circ\), and assuming \(g = 10\text{ m/s}^2\), \(v = \sqrt{10 \times 90 \times tan 45^\circ} = \sqrt{900 \times 1} = 30\text{ m/s}\).

At the top of a hill, "weightlessness" implies the normal force is zero. The centripetal force is provided solely by gravity: (mg = \frac{mv^2}{R}). Thus, (v = \sqrt{gR}). Using (g = 9.8\text{ m/s}^2) and (R = 20\text{ m}), (v = \sqrt{9.8 \times 20} = \sqrt{196} = 14\text{ m/s}). This speed lies between \(14\text{ m/s}\) and \(15\text{ m/s}\).

Concept: Centripetal force is provided by the tension in the string.
Formula: Centripetal force \(F_c = \frac{mv^2}{l}\). Here, \(F_c = T\).
Solution: \(T = \frac{mv^2}{l}\). Rearranging for \(v\), we get \(v^2 = \frac{Tl}{m}\), so \(v = \sqrt{\frac{Tl}{m}}\).

Concept: Understanding apparent forces in a non-inertial rotating frame of reference.
Solution: When milk is churned, the denser skim milk experiences a larger centripetal force and moves towards the outer edge, while the less dense cream moves towards the center of rotation due to centrifugal force. Thus, they separate.