two particles of masses 2kg and 3kg are moving with velocities u1=3i+4j and u2=5i respectively. what is the linear momentum of mass of 2kg with respect to center of mass system of these two particles?
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The center of mass is calculated using the formula:
COM = (m1*r1 + m2*r2) / (m1 + m2),
where m1 and m2 are the masses of the particles, and r1 and r2 are the position vectors of the particles relative to some reference point.
In this case, let's assume the reference point is the origin.
Given:
m1 = 2 kg
m2 = 3 kg
u1 = 3i + 4j (velocity of the 2 kg mass)
u2 = 5i (velocity of the 3 kg mass)
To find the COM, we need to calculate the position vectors r1 and r2. Since we assumed the origin as the reference point, the position vectors are equal to the initial positions of the particles.
Let's assume the initial positions are:
r1 = 0i + 0j
r2 = 0i
Substituting these values into the COM formula, we have:
COM = (2 * (0i + 0j) + 3 * (0i)) / (2 + 3)
= (0i + 0j) / 5
= 0i + 0j
So, the center of mass of the system is at the origin (0, 0).
Next, we need to calculate the velocity of the 2 kg mass with respect to the center of mass system. This can be found using the formula:
v1_CM = u1 - V_CM,
where v1_CM is the velocity of the 2 kg mass with respect to the center of mass, u1 is the initial velocity of the 2 kg mass, and V_CM is the velocity of the center of mass.
In this case, V_CM is 0i + 0j since the center of mass is at rest.
Therefore:
v1_CM = u1 - (0i + 0j)
= 3i + 4j
The linear momentum (p) is given by the formula:
p = m1 * v1_CM,
where m1 is the mass of the 2 kg particle, and v1_CM is its velocity with respect to the center of mass.
Substituting the values:
p = 2 kg * (3i + 4j)
= 6i + 8j
So, the linear momentum of the 2 kg mass with respect to the center of mass system is 6i + 8j.