In the material, Force as a vector, we learned about the sum of forces or the resultant and learned that if the resultant force on an observed body is equal to zero, the body is in equilibrium. This means that the body is stationary or moving uniformly and in a straight line. In this material, we will ask what happens if the sum of all the forces acting on the body is not equal to zero.
If the sum of the forces on the selected body is not equal to zero, then:
the body deforms or
the speed or direction of motion of the body changes.
In this material, we will limit ourselves to the case when the speed of motion of the body changes, and we will discuss the change in the direction of motion in the material, Forces in a circular Path.
The relationship between force, mass, and acceleration is described by the law of dynamics. The law covers the following cases of the action of a force on the motion of a body:
The body is initially at rest and a force begins to act on it. The body begins to move in the direction of the force and its velocity increases uniformly with time. The motion is uniformly accelerated.
The body has an initial velocity and a force begins to act on it:
the direction of the force is the same as the direction of motion of the body. The body begins to accelerate. Its initial speed increases uniformly, the motion is uniformly accelerated.
the direction of the force is opposite to the direction of motion of the body. The body begins to brake, its initial speed decreases uniformly, the motion is uniformly decelerated.
A special case of the law of dynamics is when no force acts on the observed body. This case is identical to the case when the resultant of the forces on the observed body is equal to zero; the body then:
remains in its position, or
moves uniformly or linearly. Because of this persistence at the same position (if no force acts on the body), we say that the body persists in motion - and this particular case is called Newton's first law or the law of inertia.
Let's assume a car is at rest and we start the engine and release the clutch to start the car. The engine power is transmitted to the wheels and from the wheels to the road. The car accelerates (Figure 1). We know by feeling that the acceleration of the car will be greater if:
the power of the engine is more (produced more force);
the mass of the car is less.
Figure 1: A car accelerates
We are interested in how the quantities; force, mass, and acceleration are related to each other. Newton's 2nd law or the law of dynamics discusses the relationship among the listed quantities: Newton's second law or the law of dynamics states that the mass 
moves with uniform acceleration 
if a constant force 
acts on it. We write it mathematically as:
From the above equation, we can also express the unit for the force, 
(Newton) in terms of the base units (
). We do this by writing the unit (Newton) for force, the unit 
for mass, and the unit 
for acceleration:
The law of dynamics also applies if several forces act on a body, the resultant of which does not change with time - it is a constant. In this case, the law of dynamics states that a body moves with uniform acceleration if the sum of all forces acting on it is constant:
where:
A body moves with uniform acceleration if a constant force acts on it or if the resultant of all the forces acting on it is constant.
The acceleration acts in the direction of the force (or the resultant force).
Acceleration always has the direction of the force.
If the force acts in the same direction as the motion of the body, the acceleration is positive and the velocity of the body increases uniformly (see Figure 2, left).
If the action of the force is opposite to the motion of the body, the force is negative relative to the speed of the motion of the body. Consequently, the acceleration of the body is also negative and the body decelerates (see Figure 2, right).
Figure 2: An external force can accelerate a body (left figure) or decelerate it (right figure)
A special example of the law of dynamics is that the sum of all forces acting on a moving body of mass is zero. In this case, the acceleration is also zero. This law is also known as Newton's first law.
Let's look at Newton's first law mathematically:
We found that if the sum of the forces acting on the body is zero (
), it follows that the change in velocity is zero. The change in velocity is zero in two cases:
if the body is at rest;
if the body moves uniformly and in a straight line.
Newton's first law states that a body is at rest or moving uniformly and in a straight line if the resultant of all the forces acting on it is equal to zero.
The force of gravity (weight) or gravitational force on the planet Earth is the result of the attractive force of two masses: the planet Earth and a body (more on this in the material, Gravitational Field). At this point, it is enough to know that the Earth attracts every body on the planet. This force of attraction is called the force of gravity or the weight of a body or gravitational force. It is therefore denoted by 
of 
.
Let us apply the law of dynamics:
As we learned in the material, Vertical motion under gravity, the gravitational acceleration is not a constant but depends on:
places on Earth (it is slightly higher at the poles than at the equator);
distance from the centre of the Earth (gravitational acceleration decreases with height);
soil composition (heavy minerals increase gravitational acceleration).
The accuracy with which the gravitational acceleration is calculated depends on the other data in the given exercise. Let's list some approximations of gravitational acceleration:
The result of gravitational acceleration is that bodies fall towards the ground. However, bodies do not fall towards the ground if the force of gravity (weight) is balanced by the force of the ground (normal force).
The force of gravity (weight) is obtained by the law of dynamics by using the mass and acceleration of free fall 
.
