Calculating the Torque in a Rotating System: Understanding the Physics and Mathematics

Torque, a crucial concept in rotational mechanics, is a measure of the force that can cause an object to rotate about an axis. In this article, we will explore the process of calculating the torque about the origin for a specific system involving a stone swinging in a circle around a central axis.

Understanding the System

The problem involves a 2.0 kg stone attached to a 0.50 m long string, which is swung at a constant angular velocity of 12 rad/s. The circle is parallel to the xy-plane and centered 0.75 m from the origin along the z-axis.

Calculating Torque

To calculate the torque about the origin, we will use the formula for torque:

( mathbf{tau} mathbf{r} times mathbf{F} )

Step 1: Determine the Position Vector ( mathbf{r} )

The position vector ( mathbf{r} ) from the origin to the point where the force is applied has a magnitude of 0.50 m in the xy-plane and is shifted along the z-axis by 0.75 m. If the stone is at an angle ( theta ) from the positive x-axis, the position vector is given by:

[ mathbf{r} 0.75hat{i} - 0.50cos(theta)hat{i} 0.50sin(theta)hat{j} 0.75hat{k} ]

Step 2: Determine the Force Vector ( mathbf{F} )

The force acting on the stone is the centripetal force required to keep it moving in a circle, provided by the tension in the string. The centripetal force ( F_c ) is calculated as:

[ F_c momega^2 r ]

Where:

( m 2.0 text{ kg} ) is the mass of the stone, ( omega 12 text{ rad/s} ) is the angular velocity, ( r 0.50 text{ m} ) is the radius of the circular path.

Substituting the values, we get:

[ F_c 2.0 text{ kg} cdot 12^2 text{ rad/s}^2 cdot 0.50 text{ m} 144 text{ N} ]

The force vector ( mathbf{F} ) points towards the center of the circle. In the xy-plane, the force can be expressed as:

[ mathbf{F} -144left(cos(theta)hat{i} sin(theta)hat{j}right) ]

Step 3: Calculate the Torque

The torque ( tau ) is calculated by taking the cross product of the position vector ( mathbf{r} ) and the force vector ( mathbf{F} ).

Given that the force is directed towards the center of the circle, we can represent:

[ mathbf{F} -144left(cos(theta)hat{i} sin(theta)hat{j}right) ]

Substituting the values into the torque formula:

[ tau rF sin(90^circ) ]

Where:

( r 0.50 text{ m} ) is the distance from the z-axis to the stone, ( F 144 text{ N} ) is the centripetal force, ( phi 90^circ ) is the angle between ( mathbf{r} ) and ( mathbf{F} ).

Thus, the magnitude of the torque is:

[ tau 0.50 cdot 144 cdot sin(90^circ) 0.50 cdot 144 cdot 1 72 text{ Nm} ]

Conclusion

The magnitude of the torque about the origin is 72 Nm.

Additional Insights

This example highlights the importance of understanding rotational mechanics and how to apply vector operations to solve complex problems in physics. Understanding the principles of torque, force, and angular velocity can help in designing safer and more efficient mechanical systems.