A force is applied on a door at point P making an angle `theta ` with position vector `bar(r )` , by partical observation , give the value of theta for which on has to exert minimum force to rotate the door ?

Options

A. `0^(0)`
B. `pi`
C. `pi/2`
D. `pi/3`

Explanation

To find the angle \( \theta \) at which the minimum force is required to rotate the door, we can analyze the torque produced by the applied force. Here’s the step-by-step solution:

### Step 1: Understand the Torque Concept
Torque (\( \tau \)) is the measure of the rotational force applied to an object. It is given by the formula:
\[ \tau = r \cdot F \cdot \sin(\theta) \]
where:
- \( r \) is the distance from the pivot point (hinge) to the point of force application,
- \( F \) is the magnitude of the applied force,
- \( \theta \) is the angle between the position vector \( \vec{r} \) and the force vector \( \vec{F} \).

### Step 2: Identify Components of Force
When a force \( F \) is applied at an angle \( \theta \) to the position vector \( \vec{r} \), it can be resolved into two components:
- The component along the direction of \( \vec{r} \): \( F \cos(\theta) \)
- The component perpendicular to \( \vec{r} \): \( F \sin(\theta) \)

### Step 3: Determine the Effective Torque
The effective torque that causes the door to rotate is produced by the perpendicular component of the force:
\[ \tau = r \cdot (F \sin(\theta)) \]
To open the door easily, we need to maximize this torque.

### Step 4: Maximize the Torque
To maximize the torque, we need to maximize the \( \sin(\theta) \) term. The sine function reaches its maximum value of 1 when:
\[ \theta = 90^\circ \quad \text{(or } \frac{\pi}{2} \text{ radians)} \]

### Step 5: Conclusion
Thus, the angle \( \theta \) for which one has to exert minimum force to rotate the door is:
\[ \theta = 90^\circ \]