Rotational Equilibrium

It's the last question of a trivia competition and the host asks, " What physics concept do the Golden Gate Bridge and a ceiling fan have in common?" Your first reaction is confusion, but then you quickly yell "equilibrium", and you're correct. Ceiling fans demonstrate the concept of rotational equilibrium while the Golden Gate Bridge demonstrates static equilibrium which is a combination of rotational equilibrium and translational equilibrium. With this in mind now, let's use this article as a starting point in understanding rotational equilibrium as well as translational and static equilibrium.

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    Rotational Equilibrium Definition

    In physics, equilibrium refers to a balance of forces. Forces are actions that alter or maintain an object's state of motion.

    Rotational equilibrium is when a system has constant rotational velocity and a net torque of zero.

    However, to fully understand rotational equilibrium we must understand torque. Torque is the rotational equivalent of a force.

    Torque, \( \tau \), is a vector quantity that quantifies the turning effect of a force applied to an object and is the rotational equivalent of linear force. A resultant torque generates a resultant (non-zero) angular acceleration.

    The SI unit for torque is \( \mathrm{N\,m} \). Convention states that a counterclockwise rotation indicates positive torque, and a clockwise rotation indicates negative torque. The amount of torque applied to an object depends on the applied force and the perpendicular distance from where the force is applied, with respect to the axis of rotation.

    Rotational Equilibrium Equation

    For a system to be at rotational equilibrium, the sum of all torques acting on a system must equal zero.

    $$\sum \vec{\tau}=0.$$

    As a result of this formula, it is important to discuss the three formulas corresponding to torque.

    • Cross-Product Formula, \( \vec{\tau} = \vec{r} \times \vec{F}, \) where \(\vec{r}\) is the lever arm measured in meters \(\mathrm{m}\) and \(\vec{F}\) is the force applied measured in newtons, \(\mathrm{N}\).
    • Magnitude Formula, \( |\tau| = |\vec{r}| |\vec{F}| \sin\theta \) where \(|\vec{r}|\) is the magnitude of the lever arm, \(|\vec{F}|\) is the magnitude of the applied force, and \(\theta\) is the angle between the lever arm and the applied force.

    Rotational Equilibrium Lever arm StudySmarter

    Fig. 1- A torque diagram depicting the applied force, distance from the rotation axis, \( r \), the angle they make, and the lever arm. The dot indicates the rotational axis going out of the page.
    • Newton's Second Law Formula, \(\tau=I\alpha, \) where \(\tau\) is the magnitude of the torque applied to a body, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration of the body as a result. Note that the moment of inertia plays the role of mass when Newton's second law is written in the rotational form.

    The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.

    Rotational Equilibrium vs. Translational Equilibrium

    For equilibrium to occur, three conditions must be met. These conditions will differ slightly based on motion as objects undergo translational or rotational motion. Translational motion is one-dimensional along a straight path and corresponds to translational equilibrium.

    Translational equilibrium is a state in which the sum of all external forces acting at an objects center of mass equals zero.

    An object is in translational equilibrium if its velocity is constant, it is stationary or moving with constant velocity, and the sum of all forces acting on the object equals zero. Meaning that left- and right-hand forces are equal or up and down forces are equal. When net force equals zero, we know that the object is not accelerating according to Newton's second law, \( F=ma \).

    Rotational Equilibrium Free-body diagram StudySmarterFig. 2 - A free-body diagram of equal and opposite forces canceling one another out.

    Conversely, rotational balance corresponds to rotational motion, that is, circular motion around an axis. An object is in rotational equilibrium if its angular velocity is constant, the object is at rest or moving at a constant angular velocity, and the sum of all torques acting on the object is zero. Meaning all clockwise and counterclockwise forces are equal. When net torque equals zero, we know that the object has no angular acceleration according to Newton's second law in rotational form, \( \tau=I\alpha \).

    Rotational Equilibrium See-saw StudySmarterFig. 3 - This see-saw is in rotational equilibrium as the total torque about the equilibrium point is zero.

    Rotational Equilibrium and Static Equilibrium

    Static means stationary or at rest while equilibrium means balance.

    Static equilibrium is the balanced state of rest that objects can occupy when no net force or torque is acting on the object.

    When an object is in static equilibrium, it is at rest. The object is not moving along an axis or rotating about an axis. The object is completely stationary. Two conditions are required for an object to be in static equilibrium. \( \text{Condition 1} \) refers to translational equilibrium and states that all forces acting on the object along any axis must equal zero. This means that

    \begin{align} \sum{F_x}=0, \\\sum{F_y}=0, \\\sum{F_z}=0. \\\end{align}

    \( \text{Condition 2} \) refers to rotational equilibrium and states that all torques acting on the object along any axis must equal zero. This means that

    \begin{align}\sum{\tau_x}=0, \\ \sum{\tau_y}=0, \\ \sum{\tau_z}=0.\\\end{align}

    If these conditions are satisfied, objects are said to be in static equilibrium.

    Real-Life Applications of Static Equilibrium

    Static equilibrium is an extremely important concept when building suspension bridges. Take the Golden Gate Bridge for example. During construction, engineers put a lot of work into determining the forces at every point in the bridge. This allows them to determine how strong the suspension cables must be along the bridge. Cables must be strong enough to suspend the bridge while not being too heavy. Static equilibrium enables them to find the forces acting on the bridge at any point and in turn, select the proper cable and/or other materials needed to successfully construct the bridge. This is good for us because I for one would not want to drive across a moving bridge, would you?

    Rotational Equilibrium Examples

    Using our newfound knowledge of rotational equilibrium, let's complete two examples to help solidify this concept.

    Two children are sitting on a stationary seesaw, each is located \( 1.5\,\mathrm{m} \) from the center. If a force of \( 588\,\mathrm{N} \) is exerted on the seesaw by one child, calculate the force exerted by the other child so that the seesaw is kept in equilibrium.

    Solution

    For an object to be in equilibrium, we know that all torques acting on the object must equal zero. Therefore, if the distance from the pivot is the same on each side and a force of \( 588\,\mathrm{N} \) is applied on one side, we know that an equal force of \( 588\,\mathrm{N} \) must be applied on the opposite side to satisfy the conditions of rotational equilibrium.

    \[\begin{align}\tau_1+\tau_2&=0,\\r_1F_1\sin\theta+r_2F_2\sin\theta&=0.\\\end{align} \]

    For \( \theta \), we know that both angles are 90 degrees due to the force of gravity forming right angles with each side of the beam. Since \( \sin(90^\circ)=1 \), we can rewrite the equation as \(r_1 F_1+r_2 F_2=0\).

    \[ \begin{align}r_1F_1+r_2F_2&=0,\\-r_1F_2&=r_2F_2,\\-\frac{r_1{F_1}}{r_2}&=F_2,\\-\frac{(1.5\,\mathrm{m})(588\,\mathrm{N})}{1.5 \,\mathrm{m}}&=F_2,\\F_2&=588\,\mathrm{N}.\\\end{align} \]

    Thus the second child is sitting on the other side of the equilibrium point, indicated by the negative sign.

    Let's try a slightly more complex example.

    A beam, balanced on a pin, has a \( 0.050\,\mathrm{kg}\) mass placed \( 0.32\,\mathrm{m}\) to the left of the center of the beam. Where should a \( 0.067\,\mathrm{kg}\) mass be placed, right of the center of the beam, so that the beam stays in equilibrium? Note that gravity is pushing down on the beam at the center point.

    Rotational Equilibrium Balance Beam StudySmarterFig. 4- A diagram depicting masses balanced on a beam.

    Solution

    Step 1: First, we need to determine all of the forces acting on the beam and each force's distance from the point of rotation. Remember that the force that each mass exerts on the beam is equal to the weight of the mass. Weight is calculated using the equation, \( F=mg\), where \(m\) is mass measured in \( \mathrm{kg} \) and the \(g\) is acceleration due to gravity which is a constant, \( 9.8\,\mathrm{\frac{m}{s^2}}\). Therefore, our calculations are as follows.

    \[ \begin{align} F_1&=m_1{g},\\F_1&=(0.050 \,\mathrm{kg})\left(9.8\,\mathrm{\frac{m}{s}^2}\right),\\F_1&=0.49 \,\mathrm{N}. \\\end{align} \]

    For the second mass, we have

    \[ \begin{align} F_2&=m_2{g},\\F_2&=(0.067 \,\mathrm{kg})\left(9.8\,\mathrm{\frac{m}{s}^2}\right),\\F_2&=0.66 \,\mathrm{N}. \\\end{align}\]

    Now, we know that the \( F_1 \) has a corresponding radius of \( 0.32\,\mathrm{m} \) as this information is given to us. However, we must determine the corresponding radius for \( F_2\).

    Step 2: To determine the radius corresponding to \( F_2 \), we need to calculate the sum of all torques acting on the beam. Remember that for an object to be in equilibrium, the sum of all torques must be zero. However, before proceeding further, we need to identify which torque formula applies. The formula

    \( \tau=rF\sin\theta \) applies and allows us to write the following.

    \[ \begin{align}\tau_1+\tau_2&=0,\\r_1F_1\sin\theta+r_2F_2\sin\theta&=0.\\\end{align}\]

    For \( \theta \), we know that both angles are 90 degrees due to the force of gravity forming right angles with each side of the beam. Furthermore, we know that the first torque is positive because of the right-hand rule. Pointing our fingers to the left and curling them down in the direction of gravity results in our thumb pointing outward. For the second torque, pointing our fingers to the right and curling them downward results in our thumb pointing inward, indicating a negative torque. Since \( \sin \left(90^{\circ}\right)=1\) we can simplify this equation \(r_1 F_1+r_2 F_2=0\).

    Step 3: Rearrange our equation,\(r_1 F_1+r_2 F_2=0\), in terms of \(r_2\) and insert our given variables. Note that we are taking the positive direction to the right of the beam, resulting in

    \[ \begin{align}r_1F_1+r_2F_2&=0,\\r_1F_2&=-r_2F_2,\\-\frac{r_1{F_1}}{F_2}&=r_2,\\-\frac{(-0.32\,\mathrm{m})(0.49\,\mathrm{N})}{0.66 \,\mathrm{N}}&=r_2,\\r_2&=0.24\,\mathrm{m}.\\\end{align} \]

    The second mass should be placed \( 0.24\,\mathrm{m}\) to the right of the center of the beam.

    Rotational Equilibrium - Key takeaways

    • Rotational equilibrium is a state in which neither a system's state of motion nor its internal energy state changes with respect to time.
    • Torque, \( \tau \), is a vector quantity that quantifies the turning effect of a force applied to an object.
    • For a system to be at rotational equilibrium, the sum of all torques acting on a system must equal zero, \( \sum \tau=0 \).
    • Translational equilibrium is a state in which the sum of all external forces acting on an object equals zero.
    • Static equilibrium is the state of complete rest.

    References

    1. Fig. 1 - Torque diagram, StudySmarter Originals.
    2. Fig. 2 - Free body diagram, StudySmarter Originals.
    3. Fig. 3 - See-saw in equilibrium, StudySmarter Originals.
    4. Fig. 4 - Balance beam, StudySmarter Originals.
    Frequently Asked Questions about Rotational Equilibrium

    What is rotational equilibrium?  

    Rotational equilibrium is when a system has constant rotational velocity and a net torque of zero.

    What is an example of rotational equilibrium? 

    An example of rotational equilibrium is when two children sitting on a stationary seesaw where each exerts the same amount of force on equal sides of the seesaw.

    What is the formula for rotational equilibrium? 

    For a system to be at rotational equilibrium, the sum of all torques acting on a system must equal zero. The formula is the sum of all torques equals zero.

    What are conditions for rotational equilibrium? 

    An object is in rotational equilibrium if its angular velocity is constant, the object is at rest or moving at a constant angular velocity, and the sum of all torques acting on the object is zero. 

    What are the 3 types of equilibrium? 

    The three types of equilibrium are rotational, translational, and static.

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