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: $\tau =\overrightarrow{r}\times \overrightarrow{F}$ where $\tau$ is the torque, $\overrightarrow{r}$ is the position vector, and $\overrightarrow{F}$ 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 $\overrightarrow{r}$ and $\overrightarrow{F}$.

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) ${90}^{\circ }$

Key concepts

Angle between force and position vector

The angle between the force and the position vector, often represented by $\theta$, plays a critical role in determining how much torque is generated. Torque, also known as the moment of force, is directly affected by this angle. When we talk about the angle between two vectors, we are describing how "off-kilter" or perpendicular these vectors are to each other. Imagine the position vector as a lever or arm, and the force acting on it as a push or pull.
Normally, the torque is calculated using the formula $|\tau |=rF\sin \theta$, where $r$ is the length of the position vector and $F$ is the magnitude of the force. The term $\sin \theta$ tells us how much of the force is actually contributing to the rotational effect, or torque. When $\theta$ is small, close to $0^{\circ }$, the sine of the angle is also small, and very little torque is produced.
Understanding this relationship between angle and torque can help explain why rotating objects might spin faster or slower depending on where a force is applied.

Maximum torque

The concept of maximum torque is crucial in understanding how efficiently we can use a force to cause rotation. To find the angle that maximizes torque, we focus on the mathematical behavior of the sine function in the torque equation $|\tau |=rF\sin \theta$. This equation tells us that torque depends not just on the force and the position vector, but also on the angle between them.
Maximum torque occurs when the sine function reaches its highest value, which is $1$. This happens at an angle of ${90}^{\circ }$. Thus, when the force vector and position vector are perpendicular to each other, the torque is maximized. It's like pushing a door right at the edge, perpendicular to the hinges, to make it swing effortlessly.
By understanding this concept, one can strategically apply forces in engineering and physics problems to get the desired rotational effect with the least amount of effort.

Sine function property

The sine function has some unique characteristics that affect how torque behaves in rotational systems. When studying torque, the sine function appears because it models how the angle between force and position vector influences the rotational effect. The formula $\sin \theta$ ranges from $-1$ to $1$, with critical points at specific angles.
The most important property, in this context, is that $\sin \theta$ reaches its maximum value of $1$ when $\theta ={90}^{\circ }$. This means that the entire magnitude of the force is contributing to the torque. Conversely, at $0^{\circ }$ or ${180}^{\circ }$, when $\sin \theta$ is $0$, no torque is produced, regardless of the force magnitude.
Understanding this property helps in predicting the behavior of systems where rotational movement and forces are involved. It's why you push perpendicularly to open a door; the sine function informs this efficient application of force.

Moment of force

The moment of force is also known as torque, and it's a key concept in mechanics, especially when discussing how forces cause objects to rotate. The formula for torque is $\tau =\overrightarrow{r}\times \overrightarrow{F}$, which involves a vector product, evident in the cross product notation.
The magnitude of the torque can be expressed as $|\tau |=rF\sin \theta$. Here, $\overrightarrow{r}$ is the position vector from the pivot point to the point where the force $\overrightarrow{F}$ is applied. This equation highlights the importance of not just the force magnitude and direction, but also where and how the force is applied along the position vector.
Torque leads to rotational motion, a fact that is used in various applications such as engines, levers, and gears. Understanding the moment of force allows one to solve real-world problems involving rotational dynamics and mechanical advantage.