A particle moves in the `xy` plane under the influence of a force such that its linear momentum is `vecP(t) = A [haticos(kt)-hatjsin(kt)]`, where `A` and `k` are constants. The angle between the force and momentum is

Options

A. `0^(@)`
B. `30^(@)`
C. `45^(@)`
D. `90^(@)`

Explanation

To find the angle between the force and momentum of a particle moving in the xy-plane, we start with the given expression for linear momentum:

\[ \vec{P}(t) = A \left( \hat{i} \cos(kt) - \hat{j} \sin(kt) \right) \]

### Step 1: Differentiate the momentum to find the force
The force \(\vec{F}(t)\) acting on the particle is given by the time derivative of momentum:

\[ \vec{F}(t) = \frac{d\vec{P}}{dt} \]

Calculating the derivative:

\[ \vec{F}(t) = \frac{d}{dt} \left[ A \left( \hat{i} \cos(kt) - \hat{j} \sin(kt) \right) \right] \]

Using the chain rule, we differentiate each component:

\[ \vec{F}(t) = A \left( -k \hat{i} \sin(kt) - k \hat{j} \cos(kt) \right) \]

This simplifies to:

\[ \vec{F}(t) = -Ak \left( \hat{i} \sin(kt) + \hat{j} \cos(kt) \right) \]

### Step 2: Calculate the angle between force and momentum
To find the angle \(\theta\) between the vectors \(\vec{F}\) and \(\vec{P}\), we use the dot product formula:

\[ \vec{F} \cdot \vec{P} = |\vec{F}| |\vec{P}| \cos(\theta) \]

### Step 3: Calculate the dot product \(\vec{F} \cdot \vec{P}\)
Calculating the dot product:

\[ \vec{F} \cdot \vec{P} = \left( -Ak \left( \hat{i} \sin(kt) + \hat{j} \cos(kt) \right) \right) \cdot \left( A \left( \hat{i} \cos(kt) - \hat{j} \sin(kt) \right) \right) \]

Expanding this:

\[ = -Ak \left( A \left( \sin(kt) \cos(kt) - \cos(kt) \sin(kt) \right) \right) \]

Notice that the terms cancel out:

\[ = -Ak \cdot A \cdot 0 = 0 \]

### Step 4: Determine the angle \(\theta\)
Since the dot product \(\vec{F} \cdot \vec{P} = 0\), we have:

\[ |\vec{F}| |\vec{P}| \cos(\theta) = 0 \]

This implies:

\[ \cos(\theta) = 0 \]

Thus, the angle \(\theta\) is:

\[ \theta = 90^\circ \]

### Final Answer
The angle between the force and momentum is \(90^\circ\).

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