2.0 kg particle-like object moves in a plane with velocity components vx = 30 m/s and vy = 35 m/s as it passes through the point with (x, y) coordinates of (3.0, -4.0) m.
(a) What is its angular momentum relative to the origin at this moment?
kg·m2/s k
(b) What is its angular momentum relative to the point (-2.0, -2.0) m at this same moment?
kg·m2/s k
I got part a) but how do i do part b) ?
To find the angular momentum of a particle-like object, we need to use the formula:
L = m * (r x v)
where:
L is the angular momentum,
m is the mass of the object,
r is the position vector from the center of rotation (origin or any other point) to the object, and
v is the velocity vector of the object.
Let's calculate the angular momentum in both cases.
(a) Angular momentum relative to the origin:
Given:
m = 2.0 kg
vx = 30 m/s
vy = 35 m/s
Coordinates = (3.0, -4.0) m
First, we need to calculate the position vector (r):
Since the object is at coordinates (3.0, -4.0), the position vector, r, can be calculated by subtracting the coordinates of the origin (0, 0) from the object's coordinates:
rx = 3.0 m - 0 m = 3.0 m
ry = -4.0 m - 0 m = -4.0 m
Now, we can calculate the angular momentum (L):
L = m * (r x v)
Since we're working in a two-dimensional plane, the cross product (r x v) will only have a z-component, given by: (rx * vy - ry * vx)
L = m * (rx * vy - ry * vx)
= 2.0 kg * (3.0 m * 35 m/s - (-4.0 m * 30 m/s))
= 2.0 kg * (105 m^2/s + 120 m^2/s)
= 2.0 kg * 225 m^2/s
= 450 kg·m^2/s
Hence, the angular momentum relative to the origin at this moment is 450 kg·m^2/s.
(b) Angular momentum relative to the point (-2.0, -2.0) m:
Given:
m = 2.0 kg
vx = 30 m/s
vy = 35 m/s
Coordinates = (-2.0, -2.0) m
Again, we need to calculate the position vector (r):
rx = -2.0 m - 0 m = -2.0 m
ry = -2.0 m - 0 m = -2.0 m
Now, we can calculate the angular momentum (L):
L = m * (r x v)
L = m * (rx * vy - ry * vx)
= 2.0 kg * (-2.0 m * 35 m/s - (-2.0 m * 30 m/s))
= 2.0 kg * (-70 m^2/s + 60 m^2/s)
= 2.0 kg * (-10 m^2/s)
= -20 kg·m^2/s
Hence, the angular momentum relative to the point (-2.0, -2.0) m at this moment is -20 kg·m^2/s.