The equilibrium of bodies is a key concept in physics that **describes the state of a body when forces act on it in such a way that its velocity remains constant**. When we talk about balance, we often refer to two types: static balance and dynamic balance.

**Static equilibrium** occurs when a body is at rest and external forces exactly balance the effect of internal forces, without any acceleration. **Dynamic equilibrium**, on the other hand, occurs when a body moves with a constant velocity, and the forces acting on it balance so that there is no acceleration.

An interesting case of equilibrium is when **a body is placed on an inclined plane. **Balancing a body on an inclined plane requires an understanding of how forces act on an object on an inclined surface. The main forces involved in this situation are the **gravitational force, the normal force** (perpendicular to the surface of the plane) and, if there is friction, **the friction force.**

To keep the body balanced on an inclined plane, the components of the forces must be balanced so that there is no net acceleration. This may require a particular setup or the use of supports or counterweights to balance the forces.

The concept of equilibrium of bodies and the specific problem of the body on an inclined plane are fascinating topics that touch on many areas of physics, from engineering to mechanics. In this article, we will explore in detail how these forces are analyzed and balanced, with particular attention to the practical realization of balance in various situations, such as balancing a body on an inclined plane. Ready? Let’s start!

## Equilibrium conditions of a material point and a rigid body

A body is said to be in **static equilibrium** when **it is at rest and continues to stand still** .In the case of a material point, i.e. an object that is small compared to the environment in which it is located, it is at rest if the sum of all the forces is equal to 0.

A **material point** is in equilibrium when the resultant$F_{tort}$of all forces is zero, therefore equal to$0No$.But things change when we consider a rigid body, that is, an extended object that does not undergo deformations when forces are applied to it.

A **rigid body**, however, is in equilibrium when the vector sum of the applied forces is equal to$0No and$ when the vector sum of all moments of the forces applied to it is equal to$0No⋅m$.

## Barycenter or center of mass in the equilibrium of solids

The **center of gravity** , also called **center of mass** , is the geometric point corresponding to the average value of the distribution of the mass of the body, or of the system, in space. For this reason it appears to be the point of application of the force-weight of a rigid body. In the particular case of a rigid body, in fact, the center of mass has a fixed position with respect to the system.

In particular, if a body has a regular shape (e.g. sphere, cube…) its center of mass corresponds to the geometric center of the body. If a body is symmetrical and homogeneous (same density at every point), the center of gravity corresponds with its center of symmetry.

A **force** acting on a rigid body with…

- its direction
**passing through the center of gravity**causes a**translational movement**(provided there are no constraints, i.e. obstacles) - its direction, which
**does not pass through the center of gravity**, causes a**rotational movement**.

## The moment of a force

Why do we use a wrench to tighten bolts? Because the effect of the rotation is **directly proportional** not only to the strength, but also to the arm.

The **arm** of a force with respect to a point$OR$is the distance between the point$OR$and the line containing the vector$F$.

This rotation effect is a vector and is called **the moment of a force ($M$)** and is measured in$No⋅m$. The moment of a force is a vector that has:

**direction**perpendicular to the plane of rotation**outgoing direction**from the plane if the direction of rotation is anti-clockwise, incoming direction if it is clockwise**modulus**equivalent to the product of the Force ($F$), the arm and the sine of the angle between the two vectors.

This type of product is called **a vector product** . Self$θ$, the angle between the force and the arm will be 90° then the greatest effect will be obtained.

The formula can be summarized as:$M=F⋅b⋅yes no _θ$

The moment is directly proportional to the radius, therefore, to tighten a bolt, if the wrench is shorter you need to apply more force!

**Example.**

If I use a long wrench to tighten a bolt$0,2m$and I apply a force perpendicular to it of$10No$:

$M=F⋅b⋅yes no _90°=$ $10No⋅0,2m⋅1=$ $2No⋅m$

## Moment of a couple of forces in the equilibrium of solids

In the handlebars of a bicycle, the moment is given by a **pair of parallel forces ****with the same module** acting on two adjacent arms of the same length. In this case the moment is given by the sum of two equal moments and therefore double of a single moment.

This moment is called **the moment of a couple ($M_{c}$)** .

$M_{c}=2⋅F⋅b=F⋅d$

Where$d$is equal to the distance of the two parallel lines on which the Force vectors lie.

## Equilibrium of a solid on an inclined plane

When a body is placed on an inclined plane, three forces act on it:

**the weight force**($F $), which has a direction perpendicular to the ground and directed downwards**the binding force**($No$), which has a direction perpendicular to the plane and towards exiting the plane**a possible external balancing force**($F $).

Attention!

The weight force acts on the plane but downwards, if the plane is inclined it must be broken down into two components:

- The
**component parallel to the plane**($F_{//}$), with a direction parallel to the plane and towards the base - The
**component perpendicular to the plane**($F_{⊥}$), with a direction perpendicular to the plane and facing downwards.

## How to balance a solid on an inclined plane

**On an inclined plane a static sliding friction force** could act (**$F_{to}$) not negligible** , which has a direction parallel to the plane and facing upwards.

The **maximum modulus of this force** is obtained from the product between the perpendicular component of the weight force and the static sliding friction coefficient:

The frictional force is opposed by the parallel component of the weight force, which acts to push the body downwards. If the detachment friction force ($F_{p}⋅c or sα⋅μ_{s}$) is greater than the component parallel to the weight force, the body will remain in equilibrium, vice versa if the latter force is greater, then the body will move downwards pushed by a **resultant force$R$ **which is obtained thanks to the difference of the vectors$F $And$F $.

When$R$is different from [iol_placeholder type=”formula” engine=”katex” display=”inline”/] to keep the body in balance it is necessary to impart a force on it with the same modulus and the same direction but in the opposite direction, therefore upwards.

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