The rate of change of angular momentum is called

Options

A. angular velocity.
B. force.
C. torque
D. linear momentum.

Answer

To solve the question "The rate of change of angular momentum is called," we will follow these steps:

### Step 1: Define Angular Momentum
Angular momentum (L) is defined as the cross product of the position vector (R) and the linear momentum (P). Mathematically, it is expressed as:
\[ \mathbf{L} = \mathbf{R} \times \mathbf{P} \]

### Step 2: Rate of Change of Angular Momentum
To find the rate of change of angular momentum, we differentiate angular momentum with respect to time (t):
\[ \frac{d\mathbf{L}}{dt} = \frac{d}{dt}(\mathbf{R} \times \mathbf{P}) \]

### Step 3: Apply the Product Rule
Using the product rule for differentiation, we can expand this expression:
\[ \frac{d\mathbf{L}}{dt} = \frac{d\mathbf{R}}{dt} \times \mathbf{P} + \mathbf{R} \times \frac{d\mathbf{P}}{dt} \]

### Step 4: Substitute Linear Momentum
Since linear momentum (P) is defined as \( \mathbf{P} = m\mathbf{v} \) (where m is mass and v is velocity), we can express the rate of change of momentum as:
\[ \frac{d\mathbf{P}}{dt} = m \frac{d\mathbf{v}}{dt} = m\mathbf{a} \]
where \( \mathbf{a} \) is acceleration.

### Step 5: Substitute into the Equation
Now substituting back into our equation:
\[ \frac{d\mathbf{L}}{dt} = \frac{d\mathbf{R}}{dt} \times \mathbf{P} + \mathbf{R} \times m\mathbf{a} \]
Here, \( \frac{d\mathbf{R}}{dt} \) is the velocity vector \( \mathbf{v} \), so we can rewrite it as:
\[ \frac{d\mathbf{L}}{dt} = \mathbf{v} \times \mathbf{P} + \mathbf{R} \times m\mathbf{a} \]

### Step 6: Analyze the Terms
The first term \( \mathbf{v} \times \mathbf{P} \) becomes zero because the cross product of a vector with itself is zero. Thus, we focus on the second term:
\[ \frac{d\mathbf{L}}{dt} = \mathbf{R} \times m\mathbf{a} \]

### Step 7: Relate to Torque
Since \( m\mathbf{a} = \mathbf{F} \) (force), we can write:
\[ \frac{d\mathbf{L}}{dt} = \mathbf{R} \times \mathbf{F} \]
The expression \( \mathbf{R} \times \mathbf{F} \) is defined as torque (\(\tau\)):
\[ \frac{d\mathbf{L}}{dt} = \tau \]

### Conclusion
Thus, the rate of change of angular momentum is called **torque**.