A block of mass m having coefficient of friction μwith the floor F is placed at one end of the s

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A block of mass m having coefficient of friction μwith the...

A block of mass $m$ having coefficient of friction $\mu$ with the floor $F$ is placed at one end of the spring. The spring is attached to this block and a vertical shafts. The floor with the shaft is given an angular acceleration $\alpha$. Then :(a) the spring cannot elongate before $t=\sqrt{\frac{\mu g}{l \alpha^{2}}}$ (b) the spring elongates as soon as the rotation starts (c) the stored energy in the spring goes on increasing right from $t=0$ onwards (d) the maximum spring force acts at $t=\sqrt{\frac{\mu g}{l \alpha^{2}}}$

A block of mass $m$ having coefficient of friction $\mu$ with the floor $F$ is placed at one end of the spring. The spring is attached to this block and a vertical shafts. The floor with the shaft is given an angular acceleration $\alpha$. Then :(a) the spring cannot elongate before $t=\sqrt{\frac{\mu g}{l \alpha^{2}}}$
(b) the spring elongates as soon as the rotation starts
(c) the stored energy in the spring goes on increasing right from $t=0$ onwards
(d) the maximum spring force acts at $t=\sqrt{\frac{\mu g}{l \alpha^{2}}}$

Key Concepts

Angular Acceleration and Inertial Effects

Angular acceleration describes the rate at which the angular velocity of a rotating body changes over time. In a system experiencing angular acceleration, the resultant inertial forces (which may include terms proportional to time squared) evolve dynamically. This relationship is crucial for establishing when the magnitude of these inertial forces becomes sufficient to overcome static friction or when the spring begins to respond to the changing force conditions.

Hooke’s Law and Spring Dynamics

Hooke’s law relates the force exerted by a spring to its displacement from the equilibrium position, with the proportionality constant known as the spring constant. In problems where a spring is coupled with a moving object, the spring's response—its extension or compression—and the associated stored potential energy depend on the balance between external forces (like centrifugal force) and restoring forces governed by Hooke’s law.

Centrifugal Force in Rotating Systems

When a system is in rotational motion, objects within experience a force acting outward from the axis of rotation, commonly referred to as the centrifugal force in a rotating frame. This inertial effect increases with the square of the angular velocity and plays a crucial role in determining when additional forces, such as those from a spring, become significant relative to other opposing forces like friction.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact, up to a maximum threshold defined by the product of the coefficient of friction and the normal force (typically mg in a horizontal setup). In dynamics problems involving rotational motion, static friction determines when the relative motion between the object and the surface begins, influencing when additional forces (such as those in a spring) overcome the resistance to motion.

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