Two galaxy clusters have identical radii,but the galaxies in cluster A have a typical orbital velocity of 700 km/s,while the galaxies in cluster B have a typical velocity of 1,400 km/s.How do the masses of these clusters compare to each other?
A) Cluster B is half as massive as cluster A.
B) Cluster B is twice as massive as cluster A.
C) Cluster B is four times as massive as cluster A.
D) Cluster B is eight times as massive as cluster A.

Question

Quizplus · Accepted Answer

The orbital velocity of a galaxy within a cluster is related to the mass of the cluster through the equation v=√(GM/r), where v is the velocity, G is the gravitational constant, M is the mass of the cluster, and r is the radius of the cluster. If we assume that the two clusters have the same radius, we can set their orbital velocities equal to each other and solve for the ratio of their masses:

v_A = √(GM_A/r)
v_B = √(GM_B/r)
v_B/v_A = √(M_B/M_A)

We are given that v_B = 2v_A, so substituting for both velocities and solving for M_B/M_A, we get:

2 = √(M_B/M_A)
4 = M_B/M_A

Therefore, cluster B is four times as massive as cluster A.

Two Galaxy Clusters Have Identical Radii,but the Galaxies in Cluster