assume you mean
R = i - 3j
and
V = 5i + 3j
L = m * R cross V
= m *
i j k
1-3 0
5 3 0
= m[ 3 k +15 k] = 18*3 k
or 54 in z (or along k) direction
K = m * [
R = i - 3j
and
V = 5i + 3j
L = m * R cross V
= m *
i j k
1-3 0
5 3 0
= m[ 3 k +15 k] = 18*3 k
or 54 in z (or along k) direction
K = m * [
Given:
Mass of the particle, m = 3.0 kg
Velocity vector, v = (5 m/s)i + (3 m/s)j
Position vector at time t, r = (m)i - (3m)j
The angular momentum, L, is given by the cross product of the position vector and the momentum vector:
L = r x p
To find the magnitude and direction of the angular momentum, we can calculate the cross product.
Step 1: Find the momentum vector:
Momentum, p = mass x velocity
p = m * v
p = 3.0 kg * [(5 m/s)i + (3 m/s)j]
p = (15 kg路m/s)i + (9 kg路m/s)j
Step 2: Calculate the cross product:
L = r x p
L = (m)i - (3m)j x [(15 kg路m/s)i + (9 kg路m/s)j]
Using the right-hand rule for cross products, we get:
L = (9m)(15 kg路m/s)k - (m)(9 kg路m/s)k
L = (135m kg路m/s - 9mk kg路m/s)k
Step 3: Find the magnitude and direction of the angular momentum:
The magnitude of the angular momentum is given by |L| = sqrt(Lx^2 + Ly^2 + Lz^2).
|L| = sqrt[(135m kg路m/s)^2 + (-9m kg路m/s)^2]
|L| = sqrt[(18225m^2 kg^2路m^2/s^2) + (81m^2 kg^2路m^2/s^2)]
|L| = sqrt[18306m^2 kg^2路m^2/s^2]
|L| = 135m kg路m^2/s
The direction of the angular momentum is given by the unit vector, k.
Therefore, the magnitude of the angular momentum about the origin at time t is 135m kg路m^2/s, and the direction is along the z-axis or in the k-direction.
First, let's find the linear momentum vector. The linear momentum is given by the formula:
p = m * v
where p is the linear momentum vector, m is the mass, and v is the velocity vector. In this case, the mass is given as m = 3.0 kg, and the velocity vector is v = (5 m/s)i + (3 m/s)j.
So, we have:
p = (3.0 kg) * [(5 m/s)i + (3 m/s)j]
= (15 kg路m/s)i + (9 kg路m/s)j
Next, let's calculate the position vector from the origin to the point r. The position vector is simply the vector from the origin (0,0) to the point (m, -3m).
r = (m)i + (-3m)j
Now, we can calculate the cross product of the position vector and the linear momentum vector to find the angular momentum about the origin.
L = r 脳 p
To find the cross product, we can use the determinant method:
L = |i j k |
|m -3m 0 |
|15 9 0 |
Expanding the determinant, we get:
L = [( -3m )(0)i - (0)(9)i] - [(m)(0)j - (15)(0)j] + [(m)(9)k - (15)( -3m )k]
= 27m^2 k
So, the magnitude of the angular momentum about the origin at time t is 27m^2 kg路m^2/s, and the direction is along the positive z-axis (k-direction).