Have you ever played or watched a game of tug of war? If we were to observe two people in a game of tug of war that was equal and neither player was winning, we would see that even though both participants may be pulling with tremendous force, that force is balanced which means they may not actually be moving. The rope in this situation is a great example of an object in equilibrium.
Fig. 1 - Friends playing tug of war.
Objects in Equilibrium Meaning
In physics, there is a state in which all the individual forces and torques exerted upon an object are balanced. This special situation in which the net force on an object turns out to be zero is called equilibrium. In equilibrium, an object can be either at rest or moving with a constant velocity, but it is not accelerating. As you may have guessed, an object in equilibrium is obeying Newton’s first law of motion which says an object maintains its velocity unless acted upon by a net force.
An object is in equilibrium when all the forces and torques that act upon the object are balanced.
There are two types of equilibrium.
The first type is static equilibrium. An object in static equilibrium is not moving: there is no translational or rotational movement in our chosen frame of reference. The second type is dynamic equilibrium, which means the object is moving with a constant velocity. It will remain moving with that exact same velocity because there are still no net forces or torques acting on the object. Because static and dynamic equilibriums depend on the frame of reference, an object may be in static equilibrium in one frame of reference and in dynamic equilibrium in another, and vice versa.
Here are some examples of static equilibrium. Keep in mind they are examples in which linear and angular velocities are equal to zero. Later we will cover more on dynamic equilibrium.
Examples of objects in static equilibrium are:
Forces on Objects in Equilibrium
If the forces acting on an object are balanced, meaning no net force is acting on it, it is in equilibrium. Objects in equilibrium follow Newton's first law of motion: an object at rest will stay at rest, and an object in motion will stay in motion. Let's look at some types of forces that an object can experience.
A normal force is a force exerted by a surface that resists a force exerted on that surface. The normal force is both a contact force and a reaction force. This use of the word “normal” comes from mathematics and means “perpendicular”, as a normal force is always perpendicular to the surface. The normal force is the reaction force that prevents objects from overlapping.
Friction always acts in a direction that opposes motion or attempted motion. Friction is both a contact force and a reaction force. Friction is always parallel to the interface between two surfaces.
A frictional force is dependent on two things.
The normal force, or the force of the two surfaces pressed against each other
The coefficient of friction, either static or kinetic, which is dependent on the smoothness of the surfaces and any lubricants used.
Static friction happens between two surfaces when there is no relative motion between them. It increases to match any applied force to prevent the two surfaces from moving relative to each other. It is at its maximum just before the object begins to move.
Kinetic friction occurs between two surfaces when they are moving relative to each other. Kinetic friction is constant as long as the normal force stays constant.
Conditions of Objects in Equilibrium
In order for an object to be considered in equilibrium, two conditions for equilibrium must be met.
The first condition of equilibrium is called translational equilibrium. It states that the net external force on the object needs to equal zero:
$$\vec{F}_\mathrm{net}=\sum \vec{F}_\mathrm{ext}=m \vec{a}=\vec{0}\,\mathrm{N}.$$
This states that the translational acceleration of the object’s center of mass must be zero.
When an object is modeled as a particle, it is the only condition that must be satisfied. For an extended object to be in equilibrium, a second condition must be met which involves the rotational motion of the extended object.
The second condition of equilibrium is called rotational equilibrium. It states that the net external torque on the object should be zero:
$$\vec{\tau}_\mathrm{net}=\sum\vec{\tau}_\mathrm{ext}=I \vec{\alpha}=\vec{0}\,\mathrm{N\,m}.$$This means that the angular acceleration along any axis of rotation is zero. Generally, when we solve equilibrium problems, we can select the axis of rotation. Thus we can select an axis where at least one force is being exerted directly on the axis of rotation. The reason we would do this is that the lever arm would be zero for this force, and thus the torque would be zero. This way, we can forget about this one force to make our life a little easier.
Graphs of Objects in Equilibrium
When objects are in equilibrium, we know the net force is zero, so when looking at a diagram such as a free-body diagram, we look to see if the objects have a net force of zero. This means that the forces in the \(x\)-direction cancel, and the forces in the \(y\)-direction cancel. If our diagram is in more dimensions, the forces in all the extra directions cancel as well.
If we consider an object hanging from a rope off the ceiling, this means the tension in the rope would cancel the weight of the object. See the figure below for the free-body diagram.
Fig. 2 - A still-hanging suspended object is in static equilibrium.
We can see this even in dynamic equilibrium situations. If we consider a falling object, eventually we will reach terminal velocity, where the weight of the object balances the air resistance acting on the object. See the figure below for the accompanying free-body diagram.
Fig. 3 - Dynamic equilibrium of falling object.
Different diagrams display different types of equilibrium and different situations. The important thing to remember is that in any equilibrium situation the forces and torques will cancel out, and this will be reflected in the diagrams.
Examples of Resultant Force on Objects in Equilibrium
A resultant force is produced by the (vector) addition of all the forces acting on the object. Looking for the equilibrant force given a free-body diagram of an object is a very common problem that is presented to students. To bring an object into equilibrium, you add the equilibrant force to the unbalanced forces on the object.
Let's walk through an example problem
Find the equilibrant force for the force vector of \( 10\, \mathrm{N}\) north and \( 10\, \mathrm{N}\) east.
Answer
This means that you are being asked to find the force that offsets the two given forces such that it brings the object into equilibrium.
To find the net force being applied to the object, we can apply vector math as follows:
- first, line up the tip of the vector to the tail of the other vector,
- add the two vectors,
- draw a straight line from the beginning point of the first vector to the ending point of the last vector,
- the Pythagorean Theorem can be used to find the magnitude of this vector.
Fig. 4 - Chart showing FBD, resultant, and equilibrant examples.
Add a single vector to the diagram to find the equilibrant vector. This will give you a net force of zero. If your total net force is \(14\,\mathrm{N}\) northeast, then the equilibrant force should bring this back into equilibrium. This means that the equilibrant force in this case is \( 14\,\mathrm{N}\) to the southwest.
Let's walk through one more example.
Below, we will base our answers on the picture of the rake, which represents a rigid uniform rake that has a stick with a mass of \( 6.0\,\mathrm{kg}\) and a length of \( 1.0\,\mathrm{m}\). The left end of the rake rests on the ground and the right end is held in equilibrium by an upward force of \( 40\,\mathrm{N}\).
How far from the right end of the rake should the force be placed to maintain equilibrium?
Fig. 5 - Equilibrium of an upward force to a rake.Since the rake stick is uniform, we know that its center of mass is at its geometric center.
Since the rake is \( 1.0\,\mathrm{m}\) long, the center of mass of the rake, \(x_\mathrm{cm}\), is at \( 0.50\,\mathrm{m}\). The force of gravity \(F_\text{g}\) pushes down at
\begin{align}F_\mathrm{g} &= mg \\&=6.0\,\mathrm{kg}\times9.81\,\mathrm{m} / \mathrm{s}^2 \\&=60\,\mathrm{N}\end{align}
at a distance of \(0.50\,\mathrm{m}\) away from the pivot point. Thus, we have a net torque of
$$\tau = 30\,\mathrm{N\,m}.$$
We know our applied force is \(40\,\mathrm{N}\), so we just need to find the distance from the pivot to apply this force such that it exerts a torque of \(30\,\mathrm{N\,m}\) to cancel the net torque acting on the rake because of its weight. Recall that the torque is given by
$$\tau = rF\sin\theta,$$and in this case, we are applying the force perpendicular to the lever, so our sine term drops out as it is equal to exactly \(1\). This leaves us with:
\begin{align}\tau &= rF, \\\implies r &= \frac{\tau}{F} \\r &= \frac{30\,\mathrm{N\,m}}{40\,\mathrm{N}} \\r &= 0.75\,\mathrm{m}.\end{align}
This means that the force should be applied \(0.25\,\mathrm{m}\) from the right end for the rake to be in equilibrium.
Objects in Equilibrium - Key takeaways
- An object is in equilibrium when all the forces and torques that act upon the object are balanced.
- If the object in equilibrium is stationary, we call this static equilibrium
- Objects that are moving can also be in equilibrium. This is called dynamic equilibrium.
- Newton’s second law can be applied to an object in accelerated motion or in a state of equilibrium.
- Translational equilibrium is the configuration of forces such that the net force exerted on the system is zero.
- Rotational equilibrium is the configuration of forces such that the net torque exerted on the system is zero.
References
- Fig. 1 - Tug of war childhood (https://pixabay.com/photos/tug-of-war-childhood-children-play-6526675/) by xuanduongvan87 (https://pixabay.com/users/xuanduongvan87-22814888/) licensed under Public Domain.
- Fig. 2 - Static equilibrium, StudySmarter Originals.
- Fig. 3 - Dynamic equilibrium, StudySmarter Originals.
- Fig. 4 - Free-body diagram, resultant, and equilibrant, StudySmarter Originals.
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