Question

When the moment of force is maximum, then what is the angle between force and position vector of the force? (A) \(90^{\circ}\) (B) \(0^{\circ}\) (C) \(30^{\circ}\) (D) \(45^{\circ}\)

Short answer

(A) \(90^{\circ}\)

Step-by-step solution

Moment of Force Formula

The moment of force, also known as torque, can be calculated using the following formula: \( { \tau = \vec{r} \times \vec{F}} \) where \(\tau\) is the torque, \(\vec{r}\) is the position vector, and \(\vec{F}\) is the force vector. The magnitude of the torque can be expressed as: \( {|\tau| = rF \sin \theta} \) where \(||\) indicates absolute value, \(r\) is the magnitude of position vector, \(F\) is the magnitude of the force vector, and \(\theta\) is the angle between \(\vec{r}\) and \(\vec{F}\).

Finding the Maximum Torque

We want to find the angle at which the torque is maximum. To do this, we will look at how the angle (\(\theta\)) influences the torque expression: \(rF \sin \theta\). Notice that the sine function has a maximum value of 1 when the angle is \({90}^{\circ}\). In other words, the maximum possible value of the sine function is 1, which occurs when the angle is \({90}^{\circ}\).

Conclusion

Putting it all together, when \(\theta = {90}^{\circ}\), the maximum torque is obtained for any given position vector and force vector. Therefore, the answer is: (A) \(90^{\circ}\)