What is the magnitude of torque acting on a particle moving in the xy - plane about the origin if its angular momentum is `4.0sqrt(t)kgm^2s^-1` ?

Options

A. `8t^(3//2)`
B. `4/(sqrt(t))`
C. `2/(sqrt(t))`
D. `3/(2sqrt(t))`

Answer

To find the magnitude of torque acting on a particle moving in the xy-plane about the origin, given its angular momentum \( L = 4.0 \sqrt{t} \, \text{kg m}^2/\text{s} \), we can follow these steps:

### Step 1: Understand the relationship between torque and angular momentum
The torque \( \tau \) acting on a particle is related to the rate of change of its angular momentum \( L \) by the equation:
\[ \tau = \frac{dL}{dt} \]

### Step 2: Differentiate the angular momentum with respect to time
Given the angular momentum:
\[ L = 4.0 \sqrt{t} \]
we need to differentiate \( L \) with respect to \( t \).

Using the power rule for differentiation:
\[ \frac{dL}{dt} = \frac{d}{dt}(4.0 \sqrt{t}) = 4.0 \cdot \frac{1}{2} t^{-1/2} \cdot \frac{dt}{dt} = 2.0 t^{-1/2} \]
This simplifies to:
\[ \frac{dL}{dt} = \frac{2.0}{\sqrt{t}} \]

### Step 3: Substitute the result into the torque equation
Now, substituting \( \frac{dL}{dt} \) into the torque equation:
\[ \tau = \frac{dL}{dt} = \frac{2.0}{\sqrt{t}} \]

### Step 4: Conclusion
Thus, the magnitude of the torque acting on the particle at any time \( t \) is:
\[ \tau = \frac{2.0}{\sqrt{t}} \, \text{N m} \]