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What is Newton's second law?

Review your understanding of Newton's second law in this free article aligned to NGSS standards.

What is Newton's second Law?

In the world of introductory physics, Newton's second law is one of the most important laws you'll learn. It's used in almost every chapter of every physics textbook, so it's important to master this law as soon as possible.
We know objects can only accelerate if there are forces on the object. Newton's second law tells us exactly how much an object will accelerate for a given net force.
a=ΣFm
To be clear, a is the acceleration of the object, ΣF is the net force on the object, and m is the mass of the object.
Looking at the form of Newton's second law shown above, we see that the acceleration is proportional to the net force, ΣF, and is inversely proportional to the mass, m. In other words, if the net force were doubled, the acceleration of the object would be twice as large. Similarly, if the mass of the object were doubled, its acceleration would be half as large.

What does net force mean?

A force is a push or a pull, and the net force ΣF is the total force—or sum of the forces—exerted on an object. Adding vectors is a little different from adding regular numbers. When adding vectors, we must take their direction into account. The net force is the vector sum of all the forces exerted on an object.
For instance, consider the two forces of magnitude 30 N and 20 N that are exerted to the right and left respectively on the sheep shown above. If we assume rightward is the positive direction, the net force on the sheep can be found by
ΣF=30 N20 N
ΣF=10 N to the right
If there were more horizontal forces, we could find the net force by adding up all the forces to the right and subtracting all the forces to the left.
Since force is a vector, we can write Newton's second law as a=ΣFm. This shows that the direction of the total acceleration vector points in the same direction as the net force vector. In other words, if the net force ΣF points right, the acceleration a must point right.

How do we use Newton's second law?

If the problem you're analyzing has many forces in many directions, it's often easier to analyze each direction independently.
In other words, for the horizontal direction we can write
ax=ΣFxm
This shows that the acceleration ax in the horizontal direction is equal to the net force in the horizontal direction, ΣFx, divided by the mass.
Similarly, for the vertical direction we can write
ay=ΣFym
This shows that the acceleration ay in the vertical direction is equal to the net force in the vertical direction, ΣFy, divided by the mass.
When using these equations we must be careful to only plug horizontal forces into the horizontal form of Newton's second law and to plug vertical forces into the vertical form of Newton's second law. We do this because horizontal forces only affect the horizontal acceleration and vertical forces only affect the vertical acceleration. For instance, consider a hen of mass m that has forces of magnitude F1, F2, F3, and F4 exerted on it in the directions shown below.
The forces F1 and F3 affect the horizontal acceleration since they lie along the horizontal direction. Applying Newton's second law to the horizontal direction and assuming rightward is positive, we get
ax=ΣFxm=F1F3m
Similarly, the forces F2 and F4 affect the vertical acceleration since they lie along the vertical direction. Applying Newton's second law to the vertical direction and assuming upward is positive, we get
ay=ΣFym=F2F4m
Warning: A common mistake people make is to plug a vertical force into a horizontal equation, or vice versa.

What do we do when a force is directed at an angle?

When forces are directed in diagonal directions, we can still analyze the forces in each direction independently. But, diagonal forces will contribute to the acceleration in both the vertical and horizontal directions.
For instance, let's say the force F3 on the hen is now directed at an angle θ as seen below.
The force F3 will affect both the horizontal and vertical accelerations, but only the horizontal component of F3 will affect horizontal acceleration; only the vertical component of F3 will affect the vertical acceleration. So we'll break the force F3 into horizontal and vertical components as seen below.
Now we see that the force F3 can be viewed as consisting of a horizontal force F3x and a vertical force F3y.
Using trigonometry, we can find the magnitude of the horizontal component with F3x=F3cosθ. Similarly, we can find the magnitude of the vertical component with F3y=F3sinθ.
Now we can proceed as usual by plugging all horizontally directed forces into the horizontal form of Newton's second law.
ax=ΣFxm=F1F3xm=F1F3cosθm
Similarly, we can plug all vertically directed forces into the vertical form of Newton's second law.
ay=ΣFym=F2F4+F3ym=F2F4+F3sinθm

What do solved examples involving Newton's second law look like?

Example 1: Newton the turtle

A 1.2 kg turtle named Newton has four forces exerted on it as shown in the diagram below.
What is the horizontal acceleration of Newton the turtle?
What is the vertical acceleration of Newton the turtle?
To find the horizontal acceleration we'll use Newton's second law for the horizontal direction.
ax=ΣFxm(Start with Newton’s 2nd law for the horizontal direction.)
ax=(30 N)cos3022 N1.2 kg(Plug in horizontal forces with correct negative signs.)
ax=26 N22 N1.2 kg(Make sure your calculator is in degree mode, if given degrees.)
ax=3.3ms2(Calculate and celebrate!)
To find the vertical acceleration, we'll use Newton's second law for the vertical direction.
ay=ΣFym(Start with Newton’s 2nd law for the vertical direction.)
ay=16 N12 N(30 N)sin301.2 kg(Plug in vertical forces with correct negative signs.)
ay=16 N12 N15 N1.2 kg(Make sure your calculator is in degree mode if given degrees.)
ay=9.2ms2(Calculate and celebrate!)

Example 2: String cheese

A wedge of cheese is suspended at rest by two strings which exert forces of magnitude F1 and F2, as seen below. There is also a downward force of gravity on the cheese of magnitude 20 N.
What is the magnitude of the force F1?
What is the magnitude of the force F2?
We'll start by either using the horizontal or vertical version of Newton's second law. We don't know the value of any of the horizontal forces, but we do know the magnitude of one of the vertical forces—20 N. Since we know more information about the vertical direction, we'll analyze that direction first.
ay=ΣFym(Start with Newton’s 2nd law for the vertical direction.)
ay=F1sin6020 Nm(Plug in vertical forces with correct negative signs.)
0=F1sin6020 Nm(Vertical acceleration is zero since the cheese is at rest.)
0=F1sin6020 N(Multiply both sides by mass m.)
F1=20 Nsin60(Solve for F1.)
F1=23 N(Calculate and celebrate!)
Now to find the force F2, we'll use Newton's second law for the horizontal direction.
ax=ΣFxm(Use Newton’s 2nd law for the horizontal direction.)
ax=F1cos60F2m(Plug in horizontal forces with correct negative signs.)
ax=(23 N)cos60F2m(Plug in value of F1=23 N obtained in the vertical calculation.)
0=(23 N)cos60F2m(Horizontal acceleration is zero since the cheese is at rest.)
0=(23 N)cos60F2(Multiply both sides by mass m.)
F2=(23 N)cos60(Solve for F2.)
F2=11.5 N(Calculate and celebrate!)

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