91Ó°ÊÓ

Torque is a fascinating and central concept in the study of rotational motion. It's essentially a force that causes an object to rotate around an axis. In simpler terms, think of torque as the rotational equivalent of linear force.

When a torque is applied to an object, like a wheel or a seesaw, it affects how and when the object spins. The strength of this effect is determined by two main factors:

• The magnitude of the force applied.
• The distance from the axis of rotation at which the force is applied, known as the lever arm.

The equation for torque \( \tau \) is:\[\tau = r \times F \times \sin(\theta)\]Where:

• \( r \) is the lever arm distance.
• \( F \) is the force applied.
• \( \theta \) is the angle between the force and the lever arm.

Torque is measured in Newton-meters (N·m), a combination of the units for force (Newtons) and distance (meters). In this context, applying a torque of \(50 \mathrm{~N} \cdot \mathrm{m}\) implies enough force is exerted at a certain distance to influence the wheel's rotational motion.

Newton's second law for rotation is similar to his famous second law of motion for linear dynamics. It states that the change in angular momentum of an object is equal to the torque applied to it. Mathematically, it's expressed as:
\[\tau = \frac{dL}{dt}\] Where:

• \( \tau \) is the torque applied.
• \( \frac{dL}{dt} \) represents the rate of change of angular momentum \( L \).

This formula helps us understand how rotational motion changes over time when a torque is applied. It shows that the greater the torque applied to an object, the faster its angular momentum changes.

In the exercise, this equation is central to solving the problem of when the angular speed of a rotating wheel becomes zero. By applying the known torque of \(50 \mathrm{~N} \cdot \mathrm{m}\), we calculate how long it should take for the wheel's rotation to slow to a stop, starting with its initial angular momentum of \(600 \mathrm{~kg} \cdot \mathrm{m}^2/\mathrm{s}\).

Angular speed refers to how quickly an object rotates or revolves around a particular point or axis. It's the angle turned by the object in a unit of time and is typically measured in radians per second \(\text{rad/s}\).

Angular speed (\(\omega\)) conveys vital information about rotational motion, similar to how linear speed tells us how fast something moves in a straight line. Formulaically, it's given by:\[\omega = \frac{\Delta \theta}{\Delta t}\]Where:

• \( \Delta \theta \) is the change in angular displacement in radians.
• \( \Delta t \) is the time period over which the displacement happens.

When torque is applied to an object with initial angular momentum, it influences the object's angular speed by either accelerating or decelerating the rotation.

In the problem provided, the goal was to determine the time needed for the wheel's angular speed to reach zero. As torque acts against the initial direction of rotation, it gradually reduces angular momentum and, thus, the angular speed of the wheel, until it comes to a stop.