Video Transcript
Two particles of masses 2.0
kilograms and 4.0 kilograms move in uniform circles with radii of 5.0 centimeters
and 𝑅 centimeters, respectively. The 𝑥-coordinate of the particle
moving in the 5.0-centimeter radius circle is given by 𝑥 as a function of 𝑡 equals
5.0 cos of two 𝑡. And the 𝑦-coordinate is given by
𝑦 as a function of 𝑡 equals 5.0 sin of two 𝑡. The 𝑥-coordinate of the center of
mass of the particles is given by 𝑥 sub cm as a function of 𝑡 equals 6.0 cos of
two 𝑡. And the 𝑦-coordinate of the center
of mass of the particles is given by 𝑦 sub cm as a function of 𝑡 equals 6.0 sin of
two 𝑡. Find 𝑅.
We want to solve for the radius of
rotation of our 4.0-kilogram particle. And to do that, we can start by
drawing a diagram of this scenario. In this example, we’re told about
two masses that are moving in circles. Mass one moves in a circle of
radius we’ve called 𝑟 of 5.0 centimeters. The second mass, mass two, moves in
a circle of radius capital 𝑅 that we want to solve for.
With respect to the smaller mass,
we’re told the 𝑥- and 𝑦-coordinates of that mass as a function of time. And we’re also told the 𝑥- and
𝑦-coordinates of the center of mass of this system of two masses as a function of
time. Even with the center of mass that
changes with time, the relationship that the center of mass is equal to the sum of
the product of each mass element multiplied by its distance from the center of mass
divided by the sum of the overall masses still applies to our situation.
Let’s look for a moment at the
𝑥-position of our particle and the 𝑥-position of the center of mass of the
system. We notice that these two
expressions have the same phase. And if we look at 𝑦 as a function
of 𝑡 and the center of mass 𝑦-coordinate, we see the same relationship that the
phase is the same. This means we can solve for the
radius of our second particle, capital 𝑅, by looking at either the 𝑥-coordinate or
separately the 𝑦-coordinate. Both will give us the same
information.
Just to choose one of the
coordinates, let’s write down the center of mass relationship in the
𝑥-direction. The 𝑥-coordinate of our system of
masses center of mass is equal to 𝑚 one times 𝑥 as a function of 𝑡 times 𝑚 two
times the function we’ve called 𝑥 sub two as a function of 𝑡, currently unknown,
all divided by the sum of 𝑚 one and 𝑚 two. Let’s plug in to this equation what
we know. We know 𝑥 sub cm, 𝑚 one, 𝑚 two,
and 𝑥 as a function of time.
When we write in all the
information we know, we see that the final form of our center of mass has an
expression cos of two 𝑡. We see that same form in the
expression of the exposition of our smaller particle. And knowing that the 5.0 that
precedes that phase information represents the radius of our smaller particle’s
rotation. This gives us a clue that the
expression for 𝑥 sub two as a function of 𝑡 will be the radius of the larger
particle, capital 𝑅, multiplied by the same phase, cos of two 𝑡.
We know that 𝑥 sub two of 𝑡 will
have this form because this is the exposition of the particle with a radius capital
𝑅. And we know the phase relationship
of this second particle’s position must be consistent with the phase relationship of
the overall center of mass of the system. So substituting this expression
into our center of mass equation, we see that the units of kilograms cancel out from
this expression. And if we multiply both sides of
our equation by 6.0 and then subtract 10.0 times the cos of two 𝑡 from both sides
and then seeing the cos of two 𝑡 appearing on both sides canceling that
trigonometric phase term out. We see that 26 is equal to 4.0𝑅 or
𝑅 is equal to 26 divided by 4.0, where 𝑅 implicitly is in units of
centimeters. To two significant figures as a
decimal, 𝑅 is equal to 6.5 centimeters. That’s the radius of the circle in
which the larger particle moves.
