A system consists of two particles. Particle 1 with mass 2.0 kg is located at (2.0 m, 6.0 m) and has a velocity of (3.1 m/s, 2.6 m/s). Particle 2 with mass 4.5 kg is located at (4.0 m, 1.0 m) and has a velocity of (1.1 m/s, 0.6 m/s). Determine the position and the velocity of the center of mass of the system.
To determine the position and velocity of the center of mass of the system, we need to use the formulas for calculating the center of mass.
The position of the center of mass (x_cm, y_cm) is calculated using the following formula:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
y_cm = (m1 * y1 + m2 * y2) / (m1 + m2)
where,
m1 and m2 are the masses of particles 1 and 2 respectively,
(x1, y1) and (x2, y2) are the positions of particles 1 and 2 respectively.
Using the given values:
m1 = 2.0 kg, x1 = 2.0 m, y1 = 6.0 m (for particle 1)
m2 = 4.5 kg, x2 = 4.0 m, y2 = 1.0 m (for particle 2)
We can substitute these values into the formulas to find the position of the center of mass:
x_cm = (2.0 kg * 2.0 m + 4.5 kg * 4.0 m) / (2.0 kg + 4.5 kg)
y_cm = (2.0 kg * 6.0 m + 4.5 kg * 1.0 m) / (2.0 kg + 4.5 kg)
Simplifying the equations gives:
x_cm = (4.0 kg*m + 18.0 kg*m) / 6.5 kg
x_cm = 22.0 kg*m / 6.5 kg
x_cm = 3.3846 m
y_cm = (12.0 kg*m + 4.5 kg*m) / 6.5 kg
y_cm = 16.5 kg*m / 6.5 kg
y_cm = 2.5385 m
Therefore, the position of the center of mass of the system is (3.3846 m, 2.5385 m).
To calculate the velocity of the center of mass, we can use the formula:
velocity_cm = (m1 * velocity1 + m2 * velocity2) / (m1 + m2)
Substituting the known values:
m1 = 2.0 kg, velocity1 = (3.1 m/s, 2.6 m/s) (for particle 1)
m2 = 4.5 kg, velocity2 = (1.1 m/s, 0.6 m/s) (for particle 2)
velocity_cm = (2.0 kg * (3.1 m/s, 2.6 m/s) + 4.5 kg * (1.1 m/s, 0.6 m/s)) / (2.0 kg + 4.5 kg)
Simplifying the equation gives:
velocity_cm = (6.2 kg*m/s, 5.2 kg*m/s) + (4.95 kg*m/s, 2.7 kg*m/s) / 6.5 kg
velocity_cm = (11.15 kg*m/s, 7.9 kg*m/s) / 6.5 kg
velocity_cm = (1.7154 m/s, 1.2154 m/s)
Therefore, the velocity of the center of mass of the system is (1.7154 m/s, 1.2154 m/s).
To find the position and velocity of the center of mass of the system, we can use the following formulas:
Position of the Center of Mass:
Xcm = (m1*x1 + m2*x2) / (m1 + m2)
Ycm = (m1*y1 + m2*y2) / (m1 + m2)
Velocity of the Center of Mass:
Vcm_x = (m1*v1_x + m2*v2_x) / (m1 + m2)
Vcm_y = (m1*v1_y + m2*v2_y) / (m1 + m2)
Given the following data:
m1 = 2.0 kg (mass of particle 1)
m2 = 4.5 kg (mass of particle 2)
x1 = 2.0 m (x-coordinate of particle 1)
y1 = 6.0 m (y-coordinate of particle 1)
v1_x = 3.1 m/s (x-velocity of particle 1)
v1_y = 2.6 m/s (y-velocity of particle 1)
x2 = 4.0 m (x-coordinate of particle 2)
y2 = 1.0 m (y-coordinate of particle 2)
v2_x = 1.1 m/s (x-velocity of particle 2)
v2_y = 0.6 m/s (y-velocity of particle 2)
Let's calculate the position and velocity of the center of mass step-by-step:
Step 1: Calculate the position of the center of mass (Xcm, Ycm):
Xcm = (m1*x1 + m2*x2) / (m1 + m2)
= (2.0 kg * 2.0 m + 4.5 kg * 4.0 m) / (2.0 kg + 4.5 kg)
= (4.0 kg*m + 18.0 kg*m) / 6.5 kg
= 22.0 kg*m / 6.5 kg
= 3.38 m
Ycm = (m1*y1 + m2*y2) / (m1 + m2)
= (2.0 kg * 6.0 m + 4.5 kg * 1.0 m) / (2.0 kg + 4.5 kg)
= (12.0 kg*m + 4.5 kg*m) / 6.5 kg
= 16.5 kg*m / 6.5 kg
= 2.54 m
Therefore, the position of the center of mass is (3.38 m, 2.54 m).
Step 2: Calculate the velocity of the center of mass (Vcm_x, Vcm_y):
Vcm_x = (m1*v1_x + m2*v2_x) / (m1 + m2)
= (2.0 kg * 3.1 m/s + 4.5 kg * 1.1 m/s) / (2.0 kg + 4.5 kg)
= (6.2 kg*m/s + 4.95 kg*m/s) / 6.5 kg
= 11.15 kg*m/s / 6.5 kg
≈ 1.715 m/s
Vcm_y = (m1*v1_y + m2*v2_y) / (m1 + m2)
= (2.0 kg * 2.6 m/s + 4.5 kg * 0.6 m/s) / (2.0 kg + 4.5 kg)
= (5.2 kg*m/s + 2.7 kg*m/s) / 6.5 kg
= 7.9 kg*m/s / 6.5 kg
≈ 1.215 m/s
Therefore, the velocity of the center of mass is (1.715 m/s, 1.215 m/s).
To summarize:
The position of the center of mass is (3.38 m, 2.54 m).
The velocity of the center of mass is (1.715 m/s, 1.215 m/s).