Video Transcript
A metallic ball moves in a straight
line with a constant velocity of magnitude one meter per second. It enters a dusty medium. If the force acting on the ball at
any instant is 10 dynes, find the rate of change of mass of the ball due to the dust
adherence to its surface.
The first thing to consider here is
what we’ve been asked to find, and that is the rate of change of mass of the
ball. This implies that the ball does not
have a constant mass; hence, we’ll need to use Newton’s second law for variable
masses. Newton’s second law for variable
masses tells us that 𝐹 is equal to 𝑚d𝑣 by d𝑡 plus 𝑣d𝑚 by d𝑡. What we have been asked to find is
d𝑚 by d𝑡. Let’s consider the pieces of
information that the question has given us.
We know that the velocity of the
ball is one meter per second, and this is a constant velocity. Therefore, d𝑣 by d𝑡 will be equal
to zero. We’ve also been given that the
force acting on the ball is 10 dynes. In order to make our calculations
easier, we need to convert this to newtons. We have that one dyne is equal to
10 to the negative five newtons. So, our force is equal to 10
multiplied by 10 to the negative five newtons or 10 to the negative four
newtons. Now, since d𝑣 by d𝑡 is equal to
zero, the first term in our formula for finding the force will also be equal to
zero. So, in the case of this question
with the constant velocity, we have that the force 𝐹 is equal to the velocity 𝑣
multiplied by the rate of change of the mass d𝑚 by d𝑡.
Substituting in our values for 𝐹
and 𝑣, we can see that 10 to the negative four is equal to one multiplied by d𝑚 by
d𝑡. Or we could write d𝑚 by d𝑡 is
equal to 10 to the negative four. Now, let’s consider the units of
this value. Our velocity 𝑣 is in meters per
second, and our force 𝐹 is in newtons. This tells us that this rate of
change of mass will be in kilograms per second. Now, we know that one kilogram is
equal to 10 cubed grams. Therefore, we can write that d𝑚 by
d𝑡 is equal to 10 to the negative four multiplied by 10 cubed grams per second. This simplifies to 10 to the
negative one grams per second, which can also be written as d𝑚 by d𝑡 is equal to
0.1 grams per second. And this is the solution to this
question; that is, the rate of change of mass of the ball due to the dust adherence
to its surface is 0.1 gram per second.
