The torque T on a body about a given point is found to be A - L where A is a constant vector and L is angular momentum of the body about that point- From this it follows thatA- d L-d t is perpendicular to L- at all times-B- the magnitude of L- does change with time-C- L- does not change with time-D- the component of L- in the direction of A- does change with time-

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Newton's Second Law: Rotatory Version

The torque T ...

Question

The torque $\to τ$ on a body about a given point is found to be $\to A \times \to L$ where $\to A$ is a constant vector and $\to L$ is angular momentum of the body about that point. From this it follows that

\frac{d \to L}{d t}

is perpendicular to

\to L

at all times.

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the component of $\to L$ in the direction of $\to A$ does change with time.

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the magnitude of $\to L$ does change with time.

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\to L

does not change with time.

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Solution

The correct option is A $\frac{d \to L}{d t}$ is perpendicular to $\to L$ at all times.
Due to law of conservation of angular momentum, $\to L$ = constant
i.e. $\to L . \to L$ = constant
or, $\frac{d}{d t} (\to L . \to L) = 0$
or, $2 \to L . \frac{d \to L}{d t} = 0$
or, $\to L ⊥ \frac{d \to L}{d t}$
Since $τ = \to A \times \to L$
$\frac{d \to L}{d t} = \to A \times \to L$
i.e., $\frac{d \to L}{d t}$ must be perpendicular to $\to A$ as well as $\to L$ .
Further the component of $\to L$ along $\to A$ is $\frac{\to A . \to L}{A} .$ Also
$\frac{d}{d t} (\to A . \to L) = \to A . \frac{d \to L}{d t} + \to L . \frac{d \to A}{d t} = 0$ ${∵ \to A ⊥ \frac{d \to L}{d t} a n d \frac{d \to A}{d t} = \to 0}$
or, $\to A . \to L$ = constant
i.e., $\frac{\to A . \to L}{A}$ = x = constant
Since $\frac{d \to L}{d t} (o r τ)$ is perpendicular to $\to L$ , hence it cannot change magnitude of $\to L$ but can surely change direction of $\to L$ .

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τ

on a body about a given point is found to be equal to

\to A \times \to L

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\to A

is a constant vector and

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τ = \frac{d \to L}{d t}

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The torque →τ on a body about a given point is found to be →A×→L where →A is a constant vector and →L is angular momentum of the body about that point. From this it follows that

The torque $\to τ$ on a body about a given point is found to be $\to A \times \to L$ where $\to A$ is a constant vector and $\to L$ is angular momentum of the body about that point. From this it follows that