Angular momentum of a single particle moving with constant speed along circular path :

Options

A. remains same in magnitude but changes in the direction
B. is zero
C. changes in magnitude but remains same in the direction
D. remains same in magnitude and direction

Answer

To solve the problem of finding the angular momentum of a single particle moving with constant speed along a circular path, we can follow these steps:

### Step 1: Understand the Definition of Angular Momentum
Angular momentum (L) of a particle is defined as the product of its moment of inertia (I) and its angular velocity (ω). The formula is given by:
\[ L = I \cdot \omega \]

### Step 2: Identify the Moment of Inertia
For a point mass (m) moving in a circular path of radius (R), the moment of inertia (I) is given by:
\[ I = mR^2 \]

### Step 3: Relate Angular Velocity to Linear Velocity
The angular velocity (ω) can be related to the linear velocity (v) of the particle by the formula:
\[ \omega = \frac{v}{R} \]

### Step 4: Substitute Values into the Angular Momentum Formula
Now we can substitute the expressions for I and ω into the angular momentum formula:
\[ L = I \cdot \omega = (mR^2) \cdot \left(\frac{v}{R}\right) \]

### Step 5: Simplify the Expression
By simplifying the expression, we get:
\[ L = mR^2 \cdot \frac{v}{R} = m v R \]

### Step 6: Analyze the Magnitude and Direction
Since the mass (m), speed (v), and radius (R) of the particle are constant, the magnitude of angular momentum (L) remains constant.

### Step 7: Determine the Direction of Angular Momentum
The direction of angular momentum is determined by the right-hand rule. If the particle is moving in a circular path, the direction of angular momentum is along the axis perpendicular to the plane of rotation. If the motion is counterclockwise, the direction is positive along the z-axis (k-cap). If it is clockwise, the direction is negative along the z-axis (-k-cap).

### Conclusion
In conclusion, the angular momentum of a single particle moving with constant speed along a circular path remains constant in both magnitude and direction.

### Final Answer
The correct option is that the angular momentum remains the same in both magnitude and direction.

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