91Ó°ÊÓ

In circular motion, centripetal force is crucial for keeping an object moving along a curved path. It is directed towards the center of the circle and is responsible for the continuous change in the direction of the object's velocity. Without this force, an object would travel in a straight line due to inertia and leave the circular path.

For an object of mass \( m \) moving at a velocity \( v \) along a circle with radius \( r \), the centripetal force \( F_c \) is given by the equation:

• \( F_c = \frac{m v^2}{r} \)

This equation indicates that the force increases with higher speeds and smaller radii.

In the context of the exercise, the centripetal force is provided by the static friction between the button and the platform. The rotation speed, given in revolutions per minute, requires conversion into a formula-friendly radian-based speed using the relation \( 1 \text{ rev} = 2\pi \text{ rad} \). This allows us to apply the centripetal force formula accurately.

Rotational kinematics deals with the motion of objects in a circular path and the properties associated with these motions. It describes how rotational speeds change over time, much like linear kinematics describes straight-line motion.

Key variables in rotational kinematics include angular velocity (\( \omega \)), which indicates how fast an object is rotating. This is often given in revolutions per minute (rev/min) but needs conversion to radians per second for precise calculations:

• \( \omega = \text{rev/min} \times \frac{2\pi}{60} \text{ rad/s} \)

This conversion ensures that we can use the angular velocity in formulas involving rotational motion.

In the problem, rotational kinematics is used to determine the angular speeds at 40 rev/min and 60 rev/min, allowing calculation of the button's maximum radius at these speeds without slipping.

The coefficient of static friction (\( \mu_s \)) is a measure of how much frictional force is required to start moving an object resting on a surface. It is a dimensionless quantity indicating how "grippy" a surface is. Higher values suggest more frictional force resisting movement.

The frictional force \( F_f \) provided by static friction must be at least equal to the centripetal force to prevent the button from sliding. This is expressed as:

• \( F_f = \mu_s \times m \times g \)
• At the point of slipping, \( F_f = F_c \)

By equating \( F_c \) and \( F_f \), we can solve for \( \mu_s \) as:

• \( \mu_s = \frac{r \omega^2}{g} \)

In the exercise, solving for \( \mu_s \) helps determine the necessary friction to keep the button in circular motion on the rotating platform. This calculation is critical for understanding how far the button can be positioned from the axis at different speeds without slipping.