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 (2.1 m/s, 4.1 m/s). Particle 2 with mass 4.5 kg is located at (4.0 m, 1.0 m) and has a velocity of (4.6 m/s, 4.6 m/s). Determine the position and the velocity of the center of mass of the system.
I know the answers are:
Position: 3.38x + 2.54y
Velocity: 3.83x + 4.45y
How does one arrive at these answers?
m1=2 kg, x1=2m, y1 6 m, v1x =2.4 m/s, v1y =4.1 m/s,
m2 = 4.5 kg, x2=4 m, y2 = 1 m, v2x=4.5 m/s, v2y =4.6 m/s.
x(c) =(m1•x1+m2•x2)/(m1+m2) =(2•2+4.5•4)/6.5 = 3.38 m,
y(c) =(m1•y1+m2•y2)/(m1+m2) =(2•6+4.5•1)/6.5 = 2.54 m.
v(cx) =(m1•v1x+m2•v2x)/(m1+m2) =(2•2.1+4.5•4.6)/6.5 = 3.83 m/s,
v(cy) =(m1•v1y+m2•v2y)/(m1+m2) =(2•4.1+4.5•4.6)/6.5 = 4.45 m/s.
Position of C.M. 3.38x+2.54y,
Velocity of C.M. 3.83x+4.45y.
To determine the position and velocity of the center of mass of the system, we can use the formulas:
Center of Mass = (m1 * r1 + m2 * r2) / (m1 + m2)
Velocity of Center of Mass = (m1 * v1 + m2 * v2) / (m1 + m2)
Let's calculate step by step:
1. Position of Center of Mass:
Given:
Particle 1's mass (m1) = 2.0 kg, position (x1, y1) = (2.0 m, 6.0 m)
Particle 2's mass (m2) = 4.5 kg, position (x2, y2) = (4.0 m, 1.0 m)
Applying the formula, we get:
Center of Mass (x_cm, y_cm) = [(m1 * x1 + m2 * x2) / (m1 + m2), (m1 * y1 + m2 * y2) / (m1 + m2)]
Substituting the values, we have:
Center of Mass (x_cm, y_cm) = [(2.0 kg * 2.0 m + 4.5 kg * 4.0 m) / (2.0 kg + 4.5 kg), (2.0 kg * 6.0 m + 4.5 kg * 1.0 m) / (2.0 kg + 4.5 kg)]
Simplifying the equation gives:
Center of Mass (x_cm, y_cm) = [18.5 m / 6.5 kg, 15.5 m / 6.5 kg]
Center of Mass (x_cm, y_cm) = [2.846 m, 2.385 m]
Therefore, the position of the center of mass is (2.846 m, 2.385 m).
2. Velocity of Center of Mass:
Given:
Particle 1's velocity (v1x, v1y) = (2.1 m/s, 4.1 m/s)
Particle 2's velocity (v2x, v2y) = (4.6 m/s, 4.6 m/s)
Applying the formula, we get:
Velocity of Center of Mass (vx_cm, vy_cm) = [(m1 * vx1 + m2 * vx2) / (m1 + m2), (m1 * vy1 + m2 * vy2) / (m1 + m2)]
Substituting the values, we have:
Velocity of Center of Mass (vx_cm, vy_cm) = [(2.0 kg * 2.1 m/s + 4.5 kg * 4.6 m/s) / (2.0 kg + 4.5 kg), (2.0 kg * 4.1 m/s + 4.5 kg * 4.6 m/s) / (2.0 kg + 4.5 kg)]
Simplifying the equation gives:
Velocity of Center of Mass (vx_cm, vy_cm) = [21.9 m/s / 6.5 kg, 33.9 m/s / 6.5 kg]
Velocity of Center of Mass (vx_cm, vy_cm) = [3.369 m/s, 5.215 m/s]
Therefore, the velocity of the center of mass is (3.369 m/s, 5.215 m/s).
Thus, the position of the center of mass is (2.846 m, 2.385 m) and the velocity of the center of mass is (3.369 m/s, 5.215 m/s).