The angular momentum of a partielc relative to a certain...

The angular momentum of a partielc relative to a certain point $O$ varics with time as $L=a+b t^{2}$, where $a$ and $b$ are constant vectors, with $a \perp b$. Find the torque $\tau$ relative to the point $O$ acting on the particle when the angle between the vectors $\tau$ and $L$ cquals 45 .

Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of an object about a chosen reference point or axis. It is a vector quantity that depends on both the mass distribution of the object and its velocity, and it plays a central role in analyzing rotational dynamics and conservation laws in physics.

Torque

Torque quantifies the rotational effect of a force acting on a particle or system. It is defined as the time derivative of angular momentum, indicating that any change in the system's angular momentum is due to an externally applied torque. This concept is fundamental to understanding how forces generate rotational motion.

Time Derivative of a Vector Quantity

Taking the time derivative of a vector function, such as angular momentum, is a key procedure in dynamics to determine how quantities change over time. This process involves differentiating each component of the vector, which is crucial for linking physical quantities like angular momentum and torque.

Vector Relationships and Angles

Understanding the relationship between vectors, including the calculation of the angle between them, is essential in physics. The angle between vectors, determined using the dot product, helps in analyzing how the directions of different quantities (such as torque and angular momentum) relate to each other, which is important in solving rotational dynamics problems.

Transcript

0:00 Hello everyone.

00:01 Let us do the following question.

00:03 We have given in the question that angular momentum of a particle, angular momentum of a particle varies as a plus bt square, where a and b are constant, and a is perpendicular to b, where a and b are constant.

00:21 And in the question it is said that a is perpendicular to b.

00:25 We have to find the tour when theta is given as 40.

00:30 5 degree.

00:31 We have to find the tor.

00:32 As we know that value of tor can be written as dl vector by dt.

00:38 You will derivate this equation d by dt is equal to a plus bt square.

00:46 So value of this can be written as 2b into t.

00:50 This can be written as 2b into t.

00:53 Let the angle, let the angle, let the angle between tau let the angle between tau and l be 45 degree at t is equal to t0 at t is equal to t0.

01:12 Then we will write the value, then we will write the value for coos alpha is equal to tau dot l vector divided by tau the mode into l the mode.

01:30 If we will put value of cos -alpha, cos -45, this is 1 by under the root 2.

01:36 This can be written as a vector plus b vector into t -nod square and l -vector can be written as 2b into t -nodd, divided by tau under the root a -square plus b -square into t -nod -rase -to -power 4...

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