A particle of mass m= 2kg is moving with uniform speed 3(2^1/2)m/s in the XOY plane along the line y= x+6. Find the magnitude of angular momentum of the particle about the origin.

To find the magnitude of the angular momentum of a particle about a point, we can use the formula for angular momentum, which is given by the vector cross product of the position vector and the linear momentum vector. Let's break this down step by step.

Understanding the Components

First, we need to identify the relevant quantities:

• Mass (m): 2 kg
• Speed (v): 3(2 1/2) m/s, which we will convert to a decimal.
• Position (y = x + 6): This line gives us the trajectory of the particle.

Calculating the Speed

The speed of the particle is given as 3(2 1/2) m/s. Converting this to a decimal:

2 1/2 = 2.5, so:

Speed = 3 * 2.5 = 7.5 m/s.

Finding the Position Vector

Since the particle moves along the line y = x + 6, we can express its position vector in terms of a parameter. Let's denote the x-coordinate as x. Then:

Position vector, r = (x, y) = (x, x + 6).

Determining the Linear Momentum

The linear momentum p of the particle is given by:

p = m * v = 2 kg * 7.5 m/s = 15 kg·m/s.

The direction of the velocity vector can be determined from the slope of the line y = x + 6. The slope is 1, indicating that for every unit increase in x, y increases by 1 unit. Therefore, the velocity vector can be expressed as:

v = (vx, vy) = (7.5/√2, 7.5/√2) = (3.75√2, 3.75√2).

Calculating Angular Momentum

The angular momentum L about the origin is given by:

L = r × p.

We can express the position vector r in terms of x:

r = (x, x + 6, 0).

And the momentum vector p can be expressed as:

p = (15√2/2, 15√2/2, 0).

Cross Product Calculation

The cross product in three dimensions is calculated as follows:

L = r × p = |i j k|

|x x + 6 0|

|15√2/2 15√2/2 0|

Calculating the determinant gives:

L = (0 - 0)i - (0 - 15√2/2 * (x + 6))j + (x * 15√2/2 - (x + 6) * 15√2/2)k.

Thus, we can simplify this to find the magnitude of the angular momentum.

Magnitude of Angular Momentum

To find the magnitude, we can evaluate the expression we derived. The key component is the z-component of the angular momentum, which is:

Lz = 15√2/2 * (x - (x + 6)) = -15√2/2 * 6 = -45√2 kg·m²/s.

Taking the magnitude, we find:

|L| = 45√2 kg·m²/s.

In conclusion, the magnitude of the angular momentum of the particle about the origin is 45√2 kg·m²/s, which is approximately 63.64 kg·m²/s when calculated numerically.