The drawing shows three particles that are moving with different velocities. Two of the particles have mass m, and the third has a mass 2m. The third particle has a velocity of v3 = +5.1 m/s. At the instant shown, the center of mass (cm) of the three particles is at the coordinate origin. What is the velocity vcm (magnitude and direction) of the center of mass?
The velocity of the center of mass (vcm) is 0 m/s, since the center of mass is at the coordinate origin.
To find the velocity of the center of mass (vcm) of the three particles, we need to consider the masses and velocities of the individual particles.
Let's denote the masses of the two particles with mass m as m1 and m2, and the mass of the third particle with mass 2m as m3. We also need to determine the velocities of the particles.
According to the problem, the velocity of the third particle (m3) is given as v3 = +5.1 m/s.
Since we know that the center of mass of the three particles is at the coordinate origin, we can assume that the total momentum of the system is zero.
Now, let's determine the direction and magnitude of the velocity of the center of mass (vcm).
Using the principle of conservation of momentum, we can write the equation as follows:
m1*v1 + m2*v2 + m3*v3 = 0 ...(Equation 1)
Since the center of mass is at the coordinate origin, the position vectors of the individual particles add up to zero. Mathematically, this can be written as:
(m1+m2+m3)*r_cm = 0 ...(Equation 2)
Here, r_cm is the position vector of the center of mass.
Since we know that the center of mass is at the coordinate origin, the position vector of the center of mass (r_cm) is zero.
Substituting r_cm = 0 into Equation 2, we get:
(m1+m2+m3)*0 = 0
(m1+m2+m3) = 0 ...(Equation 3)
Now, we can solve Equation 1 and Equation 3 simultaneously to find vcm.
From Equation 1, we rewrite it as:
m1*v1 + m2*v2 + m3*v3 = 0
m1*v1 + m2*v2 = - m3*v3
v1 = -(m3*v3)/m1 - (m2*v2)/m1 ...(Equation 4)
Similarly, we can find the relationship between v2 and v3:
v2 = -(m3*v3)/m2 - (m1*v1)/m2 ...(Equation 5)
Substituting Equation 4 into Equation 5, we have:
v2 = -(m3*v3)/(m2) + (m3*v3)/(m1*m2) + (m2*v2)/(m1)
Now, multiplying Equation 3 by m1*m2, we get:
m1*m2*(m1 + m2 + m3) = 0
m1^2*m2 + m1*m2^2 + m1*m2*m3 = 0
Now, adding this equation to Equation 5, we can solve for v2:
v2 + m3*v3/m2 = 0
v2 = -m3*v3/m2
Substituting the value of v2 into Equation 4, we can solve for v1:
v1 = -m2*v2/m1 = -(m2*(-m3*v3/m2))/m1 = m3*v3/m1
Therefore, the velocity of the center of mass (vcm) is:
vcm = (m1*v1 + m2*v2 + m3*v3)/(m1 + m2 + m3)
= (m1*m3*v3/m1 + m2*(-m3*v3/m2) + m3*v3)/(m1 + m2 + m3)
= (2m3*v3)/(3m)
= (2/3) * 5.1 m/s
= 3.4 m/s
So, the velocity of the center of mass (vcm) is 3.4 m/s in the positive direction.
To find the velocity of the center of mass (vcm), we need to calculate the weighted average of the velocities of the particles.
First, let's assign letters to represent the velocities of the particles. Let v1 and v2 be the velocities of the particles with mass m, and let v3 be the velocity of the particle with mass 2m. We are given that v3 = +5.1 m/s.
Next, we need to consider the mass of each particle. Let's call the mass of each particle m1, m2, and m3. Since two of the particles have a mass m and the third has a mass 2m, we can assign the masses as follows: m1 = m, m2 = m, and m3 = 2m.
Now, let's use the principle of conservation of momentum to find the velocity of the center of mass. The total momentum of the system is conserved, which means the sum of the momenta before and after an interaction is the same.
The total momentum (p) of the system is given by:
p = m1 * v1 + m2 * v2 + m3 * v3
Since we're interested in the velocity of the center of mass (vcm), we need to find the weighted average of the velocities. The weight is the mass of each particle. So, the velocity of the center of mass is given by:
vcm = (m1 * v1 + m2 * v2 + m3 * v3) / (m1 + m2 + m3)
Substituting the values we have:
vcm = (m * v1 + m * v2 + 2m * 5.1 m/s) / (m + m + 2m)
Simplifying the expression:
vcm = (2m * (v1 + v2) + 2m * 5.1 m/s) / (4m)
Canceling out the common factors:
vcm = (v1 + v2 + 5.1 m/s) / 2
Therefore, the velocity of the center of mass (vcm) is given by:
vcm = (v1 + v2 + 5.1 m/s) / 2
The direction of vcm will depend on the velocities v1 and v2, which are not provided in the question.