91Ó°ÊÓ

When we discuss the conservation of angular momentum in physics, we refer to situations where no external torque affects a rotating object. Angular momentum, denoted by \( L \), is calculated as \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular speed. For a particle moving in a circular path, the equation simplifies to \( L = m r^2 \omega \), with \( m \) being mass and \( r \) the radius.
The exercise asks whether angular momentum is conserved for a block attached to a cord and rotating on a frictionless surface. Since the problem states that there is no external torque—meaning no extra forces acting from outside the system—angular momentum is conserved. A key takeaway is that internal actions like pulling the cord do not affect the conservation status, as they do not introduce any external torque. Hence, angular momentum before the radius change must equal angular momentum after the radius change.

Kinetic energy, particularly for a rotating object, is given by the formula \( KE = \frac{1}{2} I \omega^2 \). In our specific context, it simplifies to \( KE = \frac{1}{2} m r^2 \omega^2 \) for a particle like the block.
This problem involves computing initial and final kinetic energies and then determining their difference to find the change in kinetic energy. Initially, with a radius of 0.300 m and angular speed of 1.75 rad/s, we compute the initial kinetic energy. When the radius is reduced to 0.150 m and the angular speed increases to 7.00 rad/s, the equation allows us to find the final kinetic energy.
The change in kinetic energy, \( \Delta KE \), is the difference between these two values, noting the block's kinetic energy has increased due to decreased radius and increased angular speed. This change illustrates how energy can be transferred within a system as rotational attributes vary.

The work-energy principle is a fundamental concept stating that the work done on an object is equal to its change in kinetic energy. In algebraic terms, this can be represented by \( W = \Delta KE \), where \( W \) is work and \( \Delta KE \) is the change in kinetic energy.
In this exercise, the cord is pulled, which causes a change in the radius of the block’s circular path. This pulling action requires a certain amount of work to be performed. According to the work-energy principle, this work is quantified as the change in the block's kinetic energy observed through the increase in speed.
As we calculated the change in kinetic energy to be 0.012024 J, an equivalent amount of work, 0.012024 Joules, is done in pulling the cord. This demonstrates how energy input into the system through work translates directly into increases in the system's kinetic energy.

Angular speed is a measure of rotation that defines how quickly an object rotates. For an object in circular motion, the angular speed is given by \( \omega = \frac{v}{r} \), where \( v \) is the linear velocity, and \( r \) is the radius of the circle.
However, in terms of conservation of angular momentum, the new angular speed can be calculated using the relationship that the initial and final angular momenta are equal since no external torques are present. The equation for the new angular speed \( \omega_f \) becomes \( \omega_f = \omega_i \times \frac{r_i^2}{r_f^2} \). This derives from setting the initial angular momentum equal to the final angular momentum

• Initially: \( L_i = m r_i^2 \omega_i \)
• Finally: \( L_f = m r_f^2 \omega_f \)

Subsequently, substituting the known values, we find that the angular speed increases to 7.00 rad/s after the cord's length is changed, illustrating how altering the radius while keeping mass and external forces constant impacts the rotation speed.