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.

To be clear, $a$a is the acceleration of the object, $\Sigma F$\Sigma, F is the net force on the object, and $m$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, $\Sigma F$\Sigma, F, and is inversely proportional to the mass, $m$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.

A force is a push or a pull, and the net force $\Sigma F$\Sigma, F 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

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 $\vec a=\dfrac{\Sigma \vec F}{m}$a, with, vector, on top, equals, start fraction, \Sigma, F, with, vector, on top, divided by, m, end fraction. 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 $\Sigma F$\Sigma, F points right, the acceleration $a$a must point right.

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

This shows that the acceleration $a_x$a, start subscript, x, end subscript in the horizontal direction is equal to the net force in the horizontal direction, $\Sigma F_x$\Sigma, F, start subscript, x, end subscript, divided by the mass.

Similarly, for the vertical direction we can write

This shows that the acceleration $a_y$a, start subscript, y, end subscript in the vertical direction is equal to the net force in the vertical direction, $\Sigma F_y$\Sigma, F, start subscript, y, end subscript, 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 $m$m that has forces of magnitude $\redD{F_1}$start color #e84d39, F, start subscript, 1, end subscript, end color #e84d39, $\blueD{F_2}$start color #11accd, F, start subscript, 2, end subscript, end color #11accd, $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54, and $F_4$F, start subscript, 4, end subscript exerted on it in the directions shown below.

The forces $\redD{F_1}$start color #e84d39, F, start subscript, 1, end subscript, end color #e84d39 and $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 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

Similarly, the forces $\blueD{F_2}$start color #11accd, F, start subscript, 2, end subscript, end color #11accd and $F_4$F, start subscript, 4, end subscript 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

Warning: A common mistake people make is to plug a vertical force into a horizontal equation, or vice versa.

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 $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 on the hen is now directed at an angle $\theta$theta as seen below.

The force $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 will affect both the horizontal and vertical accelerations, but only the horizontal component of $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 will affect horizontal acceleration; only the vertical component of $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 will affect the vertical acceleration. So we'll break the force $\greenD {F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 into horizontal and vertical components as seen below.

Now we see that the force $\greenD{F_3}$start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54 can be viewed as consisting of a horizontal force $\greenD{F_{3x}}$start color #1fab54, F, start subscript, 3, x, end subscript, end color #1fab54 and a vertical force $\greenD{F_{3y}}$start color #1fab54, F, start subscript, 3, y, end subscript, end color #1fab54.

Using trigonometry, we can find the magnitude of the horizontal component with $\greenD {F_{3x}}=\greenD {F_{3}}\text{cos}\theta$start color #1fab54, F, start subscript, 3, x, end subscript, end color #1fab54, equals, start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54, start text, c, o, s, end text, theta. Similarly, we can find the magnitude of the vertical component with $\greenD {F_{3y}}=\greenD {F_{3}}\text{sin}\theta$start color #1fab54, F, start subscript, 3, y, end subscript, end color #1fab54, equals, start color #1fab54, F, start subscript, 3, end subscript, end color #1fab54, start text, s, i, n, end text, theta.

Now we can proceed as usual by plugging all horizontally directed forces into the horizontal form of Newton's second law.

Similarly, we can plug all vertically directed forces into the vertical form of Newton's second law.

A 1.2 kg turtle named Newton has four forces exerted on it as shown in the diagram below.

To find the horizontal acceleration we'll use Newton's second law for the horizontal direction.

To find the vertical acceleration, we'll use Newton's second law for the vertical direction.

A wedge of cheese is suspended at rest by two strings which exert forces of magnitude $F_1$F, start subscript, 1, end subscript and $\redD{F_2}$start color #e84d39, F, start subscript, 2, end subscript, end color #e84d39, as seen below. There is also a downward force of gravity on the cheese of magnitude $\greenD{20 \text{ N}}$start color #1fab54, 20, start text, space, N, end text, end color #1fab54.

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—$\greenD{20\text{ N}}$start color #1fab54, 20, start text, space, N, end text, end color #1fab54. Since we know more information about the vertical direction, we'll analyze that direction first.

Now to find the force $\redD{F_2}$start color #e84d39, F, start subscript, 2, end subscript, end color #e84d39, we'll use Newton's second law for the horizontal direction.