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Lesson Explainer: The Equilibrium of a Body on a Rough Horizontal Plane Mathematics

In this explainer, we will learn how to solve problems involving the equilibrium of a body on a rough horizontal plane.

A body that is in equilibrium on a horizontal surface has zero resultant force acting on it. Two forces act on the body: its weight, π‘Š, and the normal reaction force from the surface, 𝑁, as shown in the following figure.

According to Newton’s second law of motion, the magnitude of the net force on a body is the product of the mass and acceleration of the body. The body is at rest, so the net force on it is zero. Even when the body is at rest, gravitational force acts on it. The gravitational force on the body, π‘Š, acts vertically downward and has a magnitude given by π‘Š=π‘šπ‘”, where π‘š is the mass of the body and 𝑔 is acceleration due to gravity.

The reaction force on the body from the surface is a consequence of Newton’s third law of motion. Force 𝑁 acts normally to the surface. For a body on a horizontal surface, 𝑁 acts vertically upward. For an unsuspended body in equilibrium on a horizontal surface, π‘Š=𝑁.

For a body on a smooth surface, any net force applied horizontally on the body accelerates the body horizontally. A body on a rough surface only accelerates due to a horizontal force applied to it if the magnitude of the force is greater than that of the frictional force between the body and the surface. The frictional force acts in the opposite direction to the applied force.

Let us define the maximum frictional force that can act on a body on a rough horizontal surface, also called the limiting friction.

Definition: Limiting Friction for a Body on a Rough Horizontal Surface

The limiting friction, the magnitude of the maximum frictional force, 𝐹, between a body at rest and the surface that it rests on is given by 𝐹=πœ‡π‘, where πœ‡ is the coefficient of static friction between the body and the surface.

For a horizontal surface, 𝑁=π‘šπ‘”, and so 𝐹=π‘šπ‘”πœ‡.

Let us look at an example where the maximum applied force on a body that remains in equilibrium is determined.

Example 1: Finding the Magnitude of a Force Acting on an Object in Equilibrium on a Rough Horizontal Plane

A body weighing 25.5 N rests on a rough horizontal plane. A horizontal force acts on the body causing it to be on the point of moving. Given that the coefficient of friction between the body and the plane is 317, determine magnitude of the force.

Answer

The maximum frictional force on the body is given by 𝐹=π‘šπ‘”πœ‡.

It is the case that π‘Š=π‘šπ‘”; hence, π‘šπ‘”=25.5.N

The value of πœ‡ is 317. We have, therefore, that 𝐹=25.5ο€Ό317=4.5.N

For the body to remain in equilibrium, the applied force must have the same magnitude as the frictional force, 4.5 N.

Now, let us look at an example in which a resultant force is determined, of which the limiting friction is a component.

Example 2: Finding the Resultant Force of the Normal and Friction Forces

A body is resting on a rough horizontal plane. The coefficient of friction between the body and the plane is 0.2 and the limiting friction force that is acting on the body is 80 N. Given that 𝑅 is the resultant of the force of friction and the normal reaction force, find the magnitude of 𝑅.

Answer

The forces acting on the body are shown in the following figure.

The weight and the applied force are not mentioned, but for a body at rest on a surface, the normal reaction force on the body exists due to the weight of the body, and a frictional force on such body exists due to an applied force on the body, so the unmentioned forces must exist.

The limiting friction is the maximum frictional force that the surface exerts on the body before the body starts to move due to the applied force. The limiting friction is given by limitingfriction=πœ‡π‘, where 𝑁 is the normal reaction force on the body and πœ‡ is the coefficient of static friction between the surface and the body.

The question states that the limiting friction is 80 N and that πœ‡ is 0.2. From this, we see that the reaction force is given by 80=0.2𝑁𝑁=800.2=400.N

The force 𝑅 is stated to be the resultant of the reaction force and the limiting friction, as shown in the following figure.

The magnitude of 𝑅 can be determined using Pythagoras’s theorem: 𝑅=400+80𝑅=166400𝑅=26(6400)𝑅=80√26.N

A body on a rough surface has an angle of friction. The angle of friction is the angle between the normal reaction force on the body and the resultant of the normal reaction force and the limiting frictional force on the body.

Consider the following figure showing the forces acting on a body on a rough horizontal surface. The force 𝑅 is the resultant of the normal reaction force and the frictional force, not an additional force.

A force, 𝐹, acts horizontally on the body. For values of 𝐹 where 𝐹≀𝐹,max the frictional force on the body has a magnitude equal to 𝐹. When 𝐹=𝐹,max the frictional force magnitude equals the limiting friction; hence, 𝐹=πœ‡π‘.friction

The normal reaction force and the limiting friction act perpendicularly, as shown in the following figure.

From the figure, we see that angle πœƒ between the direction of the normal reaction force and the resultant of the normal reaction force and the frictional force has a tangent that can be defined as follows.

Definition: Angle of Friction

The angle of friction for a body on a rough horizontal surface can be determined by tanπœƒ=πœ‡π‘π‘=πœ‡, where πœ‡ is the coefficient of static friction between the body and the surface and 𝑁 is the normal reaction force on the body from the surface.

Let us look at an example where the angle of friction is determined.

Example 3: Finding the Angle of Friction

Given that the coefficient of static friction between a body and a plane is √34, what is the angle of friction? Round your answer to the nearest minute if necessary.

Answer

The angle of friction can be determined using the formula tanπœƒ=πœ‡.

The value of πœƒ is given by πœƒ=πœ‡=ο€Ώβˆš34.tantan

Rounded to the nearest minute, this gives πœƒ=2325β€².∘

Frictional force between a body and a surface only exists if a normal reaction force acts on the body. The reaction force does not have to be due to the weight of the body. Let us look at an example where the weight of the body has no effect on the reaction force on the body.

Example 4: Finding the Minimum Horizontal Force That Will Cause the Body to Be in Equilibrium When Pushed against a Rough Vertical Wall

The figure shows a body of mass 30 kg being pushed against a rough vertical wall by a horizontal force ⃑𝐹. Given that the coefficient of static friction between the body and the wall is 56, determine the minimal horizontal force ⃑𝐹 that will cause the body to be in equilibrium.

Answer

Although this question involves a body on a vertical surface, it is equivalent to modeling a body on a horizontal surface, as shown in the following figure.

The reaction force on the body from the surface acts normally to the surface, so it acts in the opposite direction to ⃑𝐹 with a magnitude equal to that of ⃑𝐹. The limiting friction on the body, 𝐿, is given by 𝐿=πœ‡π‘=56𝐹.

The weight of the body is given by π‘Š=π‘šπ‘”=30𝑔.

For the body to be in equilibrium, the magnitudes of the weight and limiting friction must be equal, so we have that 56𝐹=30𝑔𝐹=1805𝑔𝐹=36𝑔.N

No value is given in the question for 𝑔. The result can therefore be expressed using the unit of kilogram-weight as 36 kg-wt.

Let us now look at an example showing that the frictional force between a body and a rough surface can act in any direction parallel to that surface.

Example 5: Finding the Weight That Will Cause the System to be at the Point of Moving

A body of weight 79 N rests on a rough horizontal table. It is attached by a light inextensible string passing over a smooth pulley fixed at the edge of the table to a weight of 41 N hanging freely vertically below the pulley. Under these conditions, the system is on the point of moving. The body is then attached by a second inextensible string passing over a second pulley fixed at the opposite end of the table to a second weight of π‘Š N hanging freely vertically below the pulley. Determine the weight π‘Š which will cause the body to be on the point of moving.

Answer

When the 79-newton-weight body is attached to the 41-newton-weight body, the following figure shows some of the forces acting on the bodies.

The frictional force on the body must be equal to the tension in the string for the body to be in equilibrium, and the tension in the string is equal to the weight of the body suspended from the string. The frictional force of 41 N is the maximum frictional force that can exist between the body and the surface because the system is on the point of moving.

When the 79-newton-weight body is attached to the body of unknown weight, the net force on the body is in the direction of the body of unknown weight, as the 79-newton-weight body is now on the point of moving in the direction of the tension in the string due to the unknown weight. The frictional force on the 79-newton-weight body again has a magnitude of 41 N, but acts in the opposite direction to the tension in the string attached to the body of unknown weight, as shown in the following figure.

The weight of the body of unknown weight is equal to the tension in the string attached to the body. The tension in this string is equal to the sum of the tension in the string attached to the 41 N body and the frictional force on the 79 N body. The unknown weight, π‘Š, is given by π‘Š=41+41=82.N

Let us summarize what we have learned from these examples.

Key Points

  • The normal reaction force for a body on a horizontal surface has a magnitude equal to the weight of the body.
  • The frictional force between a body and a rough surface acts in the opposite direction to the net force on the body and parallel to the plane.
  • The maximum frictional force between a body and a rough surface is given by 𝐹=πœ‡π‘, where 𝑁 is the normal reaction force on the body and πœ‡ is the coefficient of static friction between the body and the surface.
  • The angle of friction for a body on a rough horizontal surface is given by πœƒ=πœ‡,tan where πœ‡ is the coefficient of static friction between the body and the surface.

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