In this explainer, we will learn how to use differentiation for Newtonβs second law of motion of a particle with variable mass.
Let us first recall Newtonβs second law of motion by defining Newtonβs second law of motion for a body of constant mass.
Definition: Newtonβs Second Law of Motion for Constant Mass
When a net force acts on a body, the body accelerates in the direction of the force. The magnitude of the acceleration depends on the magnitude of the force and on the mass of the body according to the formula where is the constant mass of the body and is the acceleration of the body.
Newtonβs second law of motion can also be expressed in terms of the rate of change of the momentum of a body. The momentum, , of a body is defined as follows.
Definition: The Momentum of a Body
The momentum of a body is given by, where is the mass of the body and is the velocity of the body.
Expressing Newtonβs second law of motion in terms of rate of change of momentum gives
The mass and velocity terms are both bracketed as either term can be time varying or constant, so the rate of change of either term may need to be considered.
If the mass of an accelerating body is constant, the rate of change of the momentum of the body is given by which is the constant mass form of Newtonβs second law of motion. Equivalently, for a body that moves uniformly when a force acts on it, the rate of change of momentum of the body is given by
Suppose that a body changes in both velocity and mass with respect to time. If a body increases in both velocity and mass, then the force acting on the body must account for both the increase in velocity and the increase in mass.
When two variables change with respect to a third variable, the product rule can be used to differentiate the product of the two variables.
How To: Using the Product Rule to Differentiate a Product
Consider the variables and that both change with respect to time. The rate at which changes with respect to time is given by
If we apply the product rule to the rate of change of momentum of a body, we find that
The force acting on a body can be defined as follows.
Definition: Newtonβs Second Law of Motion in terms of Momentum Change
When a net force acts on a body, the momentum of the body changes. The rate of change of the mass and velocity of the body depends on the magnitude of the force and on the mass and velocity of the body according to the formula where is the mass of the body and is the velocity of the body.
Let us consider an example in which a body changes mass while a force acts on it.
Example 1: Finding the Force Acting on a Body with Variable Mass Moving at a Constant Velocity
Fill in the blank: The force acting on a mass varying according to the function and moving with a constant velocity of 4 m/s is .
Answer
A force acting on a body produces a change of momentum of the body:
The body is stated to have a constant velocity, so
Hence,
The value of is stated to be 4 m/s, which can be substituted to give
For a force to act on a body to move the body with constant velocity and hence with zero acceleration, the mass of the body must change over the time that the force acts on it.
The question expresses the mass of the body as a function of time. The mass in kilograms as a function of time, , is given by the value of which does indeed increase with increasing .
Differentiating with respect to time gives
The mass of the body increases by 2 kilograms per second.
The rate of change of mass can be substituted into the formula as follows:
Therefore, a body that is moving at a constant velocity of 4 m/s that starts to increase in mass by 2 kilograms per second must be acted on by a force of 8 N to maintain that constant velocity.
The mass of the body does not affect the force required to maintain its uniform motion, only the rate of change of its mass.
Let us look at an example of a force acting on a body, where both the mass and the velocity of the body vary with time.
Example 2: Finding the Force Acting on a Body with Variable Mass at Any Time
A body moves in a straight line. At time seconds, its displacement from a fixed point is given by . Its mass varies with time such that . Write an expression for the force acting on the body at time .
Answer
Both the velocity and the mass of the body vary with time, so the force acting on the body is found using
The rate of change of mass of the body can be found by differentiating the function for the mass of the body with respect to time as follows:
The mass of the body increases by 8 kilograms per second.
The function for the rate of change of the velocity of the body with respect to time is not given by the question, but the displacement as a function of time is given. The rate of change of displacement is equal to the instantaneous velocity, , of the body, and so
As is known, we can express the rate of change of as follows:
We can use the following formula:
A time-varying force must be expressed as a function of and have an instantaneous value at all instants that the force acts. Let us look at an example in which an instantaneous value of a time-varying force is determined.
Example 3: Finding the Force Acting on a Body with Variable Mass at a Given Time
A body moves in a straight line. At time seconds, its displacement from a fixed point is given by . Its mass varies with time such that . Determine the force acting upon the body when .
Answer
To determine the force acting at an instant, the function representing the change in the force with time must be determined.
Both the velocity and the mass of the body vary with time, so the force acting on the body is found using
The rate of change of mass of the body can be found by differentiating the function for the mass of the body with respect to time as follows:
The function representing the velocity of the body is given by
Differentiating the function representing with respect to time gives us
As the product of and , , is to be differentiated with respect to time, we can use the product rule to differentiate with respect to as follows:
Substituting gives us
Let us consider Newtonβs second law of motion with variable mass when it is applied to a system where the velocity is given in terms of vectors.
Definition: Newtonβs Second Law of Motion for Variable Mass in terms of Vectors
Given a body of mass , with velocity , the force that is being applied to the body is given by where both and are vector quantities.
If we are given the vector for the displacement instead of the velocity, we can differentiate the displacement with respect to time to find the velocity since the relationship between displacement and velocity is the same in vector and scalar forms. We have that
We can now look at an example of how this formula can be used.
Example 4: Finding the Force Acting on a Body Using Newtonβs Second Law for Variable Mass with Vectors
Complete the following: An object of mass 10 g is moving in a two-dimensional plane with varying mass that increases with a rate of 5 g/s. The velocity vector for the object is given by . The horizontal component of the force acting on the object to keep it moving at this velocity is dynes.
Answer
We know that the mass of the object starts at 10 g, and it increases at a rate of 5 g/s. Therefore, we can say that
Also, since the rate of increase of mass is 5 g/s, we have that
In order to find the force that is acting on the body, we will need to use Newtonβs second law for variable mass in terms of vectors. This tells us that
In order to find , we simply differentiate with respect to . This will give us
We now have all the components we need to find . Substituting into our formula, we find
Since the units we have used so far are grams and centimetres per second, the unit for our force will be dynes. We can say that
Now, the question has asked us to find the horizontal component of this force, and we are only interested in the component. Therefore, our solution is
It is reasonable to wonder what physical process could produce the increase in the mass of a body. Suppose that a body accumulates some of the matter that it makes contact with that is part of the medium that the body travels in. The mass of the accumulated matter is added to the mass of the body.
Representing such a situation realistically would involve determining the rate at which the area of contact of the body and the medium changed as well as the rate at which the velocity of the body changed. Considering only these variables would require assuming that neither the density of the medium nor the process of accumulation of matter from the medium varied due to the velocity of the body or the area of contact of the body and the medium.
Let us consider an example in which the various processes affecting the accumulation of matter by a body are represented in a simplified way.
Example 5: Finding the Force Acting on a Body with Variable Mass at a Given Time Using Newtonβs Second Law
A ball of mass 5 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 1 g/s. Find the magnitude of the force acting on the ball at time , given that the displacement of the ball is expressed by the relation where is a unit vector in the direction of the motion and the displacement is measured in centimetres.
Answer
The force acting on the ball is given by
The ball has a mass of 5 g at , which increases at a rate of 1 g/s. The function representing the change of mass of the ball with time, , is given by
Differentiating gives us
The function representing the velocity of the body is given by
Differentiating the function representing with respect to time gives us
As the product of and , , is to be differentiated with respect to time, we can use the product rule to differentiate with respect to :
Recalling that we have
Substituting gives us
The mass is in grams and the displacement is in centimetres, and so the force calculated in newtons is multiplied by the product of the number of centimetres in a metre and the number of grams in a kilogram to give a force in newtons:
One dyne is equal to N, so the force is 287 dynes.
Let us now look at another such example.
Example 6: Finding the Rate of Change of the Mass of a Ball as It Moves Through a Dusty Medium
A metallic ball moves in a straight line with a constant velocity of magnitude 1 m/s. It enters a dusty medium. If the force acting on the ball at any instant is 10 dynes, find the rate of change of the mass of the ball due to the dust adherence to its surface.
Answer
The ball has a constant velocity. So, in the formula
The fact that the velocity is constant allows us to express the formula as
The rate of increase of the mass of the ball is directly proportional to the force acting on the ball.
The force acting on the ball is 10 dynes, where
The force is equal to the rate of change of momentum of the ball, given in newtons by
The velocity of the ball is 1 m/s, so the rate of change of momentum can be written as
Combining the value of the force with the expression for the rate of change of momentum gives us where is in kilograms (kg), the force is in newtons (N), and the velocity is in metres per second (m/s). The rate of change of mass, in grams per second (g/s), is given by
Key Points
- When a net force acts on a body of constant mass, the body accelerates in the direction of the force. The magnitude of the acceleration depends on the magnitude of the force and on the mass of the body according to the formula where is the mass of the body and is the acceleration of the body.
- The momentum of a body is given by where is the mass of the body and is the velocity of the body.
- Expressing Newtonβs second law of motion in terms of rate of change of momentum gives
Using the product rule, this can be expressed as
