General Physics Chapter 1: Rotational Motion and Kinematics PDF

# Chapter 1: Rotational Motion and Kinematics There are things around us that require us to turn to function. For example, a screw won't be held in place without tightening it using a screwdriver. Another one is when we open the doorknob, it requires us to turn the knob in order for it to open. Rot...

# Chapter 1: Rotational Motion and Kinematics There are things around us that require us to turn to function. For example, a screw won't be held in place without tightening it using a screwdriver. Another one is when we open the doorknob, it requires us to turn the knob in order for it to open. Rotation can either be fast or slow, which is in line with the concept of kinematics. For example, kinematics can be applied on how fast or slow a roller coaster should be to ensure the safety of its riders. All of these things require rotation and kinematics. This chapter will discuss the basics and the application and characteristics of torque (rotation) on the different characteristics and concepts on rotational motion and kinematics. Before proceeding in this chapter, please take note of the following: 1. Torque is not a type of force. It is produced by an external force. 2. Concepts revolving on linear kinematics are almost the same as rotational kinematics. This includes the changing of most variables into complex and more complicated ones. 3. A force does not necessarily need to be perpendicular to the length of the lever arm to produce torque. Just make sure that it should always be greater than 0°. 4. Consider always the shape and the distribution of mass for rotational inertia. 5. The center of mass is not always the axis or point of rotation of an object. 6. Any object will remain in rotational equilibrium if its center of mass is above the area of support. ## Chapter 1.1: Torque ### The Concept Torque is the ability of a force to produce rotation around an axis and it's not the force itself. Therefore, torque only exists if and only if there is rotation which is produced by an external (primarily applied) force. Torque is found in everyday lives which includes the rotation of our doorknobs, the twisting of bottlecaps to enjoy our drinks, and many more applications. To better understand the concept of torque, let us look in the illustration below: **Figure 1. A balanced seesaw** - Pivot - Length: 10 feet - Length: 10 feet - Force (weight): 50 N - Force (weight): 50 N For torque to be produced, we need to consider the length of the rotating object, and the force applied to that rotating object which must be perpendicular to it. In the figure, we can see that the kids are seated 10 ft away from the pivot and they both exert 50 N of force (weight) downwards which balances the seesaw. If one of them seats nearer/further, the seesaw won't be balanced anymore. Torque can be mathematically expressed using the length of the rotating arm and the force applied to the rotating arm. - $τ = F×L$ - $τ = Fr sin θ$ **Formula 1.1** represents the torque applied on an object with an assumption that the force applied F is perpendicular to the length of the rotating arm L. However, if the force applied is not completely perpendicular to the rotating arm, we may opt to use **formula 1.2**. Ther in formula 1.2 is equal to L in formula 1.1. **Figure 2. Parts of a Torque** - Torque - Length (L/r) - Force (applied) - Pivot Point ## Chapter 1.2: Rotational Inertia ### The Concept Pursuant to Newton's first law of motion, an object remains rotating in the absence of a net torque while a non-rotating object remains stationary. In order to change this state of motion, a torque (rotation) is required by an object. Inertia of any form is a measure of “laziness” in an object. Both linear and rotational inertia depend on mass, but the difference is, rotational inertia depends on the distribution of mass while linear inertia depends solely on mass. To better understand the concept of rotational inertia, let us investigate the illustration below: **Figure 5. Rotational inertia of a barbel at different mass concentrations** - EASY TO ROTATE - DIFFICULT TO ROTATE In Figure 5, we can see that when the mass is nearer to the center (axis of rotation), the rotational inertia of the barbel is weaker. However, when the weights are concentrated furthest from the axis of rotation, the rotational inertia is stronger which makes it more difficult to rotate the barbel. Rotational inertia is different in each object. If the mass m of an object is concentrated on a point on the object, the rotational inertia can be calculated as follows: - $I = mr^2$ Where I is the rotational inertia of the object, m is the mass of the object, and r is the perpendicular distance to the axis of rotation. However, if the mass m is concentrated in different parts of an object, the rotational inertia is calculated as follows: - $I = ∑m₁r² = m₁r² + m₂r² + … m₁r²$ **Formula 1.3** is only applicable for hollow disks and solid spherical objects. For the sake of simplicity, this reviewer will not comprehensively discuss **formula 1.4**. ## Rotational Inertia in Gymnastics _**Key Takeaway**_ The body also has its own axes of rotation. These are the longitudinal, transverse, and medial axis. These axes are at the right angles of each other and passes through the center of gravity. The rotational inertia in each axis is different. **Figure 6. The different axes of rotation in the body** - Longitudinal Axis - Medial Axis - Transverse Axis Let's discuss their differences in a table form so that it will be easy to summarize the points: | Axis of Rotation | Description | Example | |-------------------|----------------|---------| | Longitudinal | Rotational inertia is least in this axis due to the concentration of body mass - Vertical head-to-toe axis - Easiest rotation to perform - Rotational inertia is increased by extending a leg or the arms | An ice skater rotates in its longitudinal axis when going through a spin | | Transverse | Divides the body into its top and bottom part - Horizontal (side-to-side) hip-by-hip intersection - Higher rotational inertia than longitudinal | An athlete doing a backflip or somersault rotates their hips which is a rotation in the transverse axis | | Medial | Divides the body into its front and back - Vertical head-to-toe intersection - Highest rotational inertia because mass is distributed the farthest | A person doing a cartwheel is a rotation in the medial axis (the illusion that the mouvement creates and that fact that it is a horizontal movement makes it a rotation in the medial axis) | ## Rotational Inertia in Rolling Objects _**Critical Analysis**_ **Scenario 1** Imagine two cylinders with equal mass. However, one is hollow, and one is a solid cylinder. Which would you think will roll down an incline faster? Let me know if the picture below is what you're imagining **Figure 7. Two cylinders rolling down an inclined plane** If we recall the formula for rotational inertia **[1.3]** and **[1.4]**, it increases as mass becomes concentrated away from the axis of rotation. If we examine closely, for a hollow cylinder, the mass is concentrated near the outer radius and not from its center (axis of rotation). For a solid cylinder however, mass is dispersed around the cylinder and not concentrated around a specific area (there can be those that are near the axis of rotation). This makes the inertia of the solid cylinder less than the hollow one which makes it roll down faster in an incline. **Scenario 2** Imagine three solid spheres rolling on the same incline. One sphere is made up of styrofoam, the other is plastic, and the other is iron metal. Which of them will have the greatest acceleration? **Figure 8. Three spheres rolling down an inclined plane** Moment of inertia in rolling motion depends on the distribution of mass and the shape of the material (they matter more than the mass of the material). The moment of inertia is constant for every shape which suggests that no matter what material or size that objects have, if they have the same shape, they will accelerate equally down the slope. **Figure 9. Moment of inertia of different objects (varying shapes)** ## Chapter 1.3: Angular Momentum ### The Concept Angular momentum is almost similar to linear momentum. The difference is the inclusion of the radial distance r of the object to its axis of rotation. Angular momentum also abides by Newton's first law like linear momentum. It is a vector quantity having both direction and magnitude. It is mathematically expressed as follows: - $L = I × ω$ Where L is the angular momentum expressed in $kg·m^2/s$, I is the rotational inertia, and ω is the rotational velocity in $rad/s$. This formula is best when mass is distributed into different areas of an object with respect to its area of rotation. For the case of an object that is small compared with the radial distance to its axis of rotation, the angular momentum can be calculated as follows: - $L = mvr$ The inertia can be calculated by multiplying its linear momentum mv to the radial distance of an object to its axis of rotation r. This formula is best for point mass approximations. **Figure 12. An object in a constant rotating motion** Torque plays a vital role in angular momentum for it is required to change the angular momentum of an object. _**Critical Analysis**_ **Scenario 1** Imagine that you are riding a bicycle but don't pedal just yet. Hop onto the bike and observe what happens. You will find yourself struggling to balance yourself and the bicycle will just keep on falling over. **Scenario 2** Imagine that you are riding a bicycle but this time, you may now pedal. If you are a bicycle rider, you can observe that when the wheels are in motion, it is easier to balance both yourself and the bicycle. - **Answer** - When the wheels are in motion, they create a force called the gyroscopic force. This force is produced due to the principle of angular momentum. It increases the resistance of the wheels to change in direction, thus, improving overall stability than stationary wheels with no angular momentum (velocity = 0). The production of the gyroscopic force brought by the principle of angular momentum is called the gyroscopic effect. Greater torque would be needed to change the direction of the angular momentum of the moving wheels. ## Conservation of Angular Momentum _**Key Takeaway**_ The law of conservation of angular momentum is almost like linear momentum, the difference is, the conservation of angular momentum applies to rotating objects. It states that the angular momentum of a closed system is constant unless acted upon by an external torque. This means that if there is no external force to change the rotational motion of an object, the product of the rotational inertia and rotational velocity will remain the same. External forces do not conserve angular momentum, but internal forces do (such as changing the rotational inertia of the object). _**Critical Analysis**_ **Scenario 1** Imagine that you're an ice skater about to spin very fast. During your spin, you experience a slight dizziness, and you've decided to extend your arms to slow your spin. Why does extending your arms help in slowing your speed? **Figure 13. Demonstration of angular momentum conservation** Answer: If we recall the formula for angular momentum **[1.5]**, it involves the product between rotational inertia and rotational velocity. Extending your arms will increase your rotational inertia but we want to conserve angular momentum. Thus, your rotational speed will decrease as a result. ## Angular Velocity and Rotational Velocity _**Key Takeaway**_ Since rotation is involved in both concepts, we will not be utilizing anymore m/s when it comes to velocities in rotational motion. For angular velocity, we will utilize radians per second (rad/s) and for rotational velocity, we will utilize revolutions per minute (RPM) or degrees per second. To solve for angular velocity given the linear velocity of an object, we can use the formula below: - $ω = v/r$ Where v is the linear velocity of the rotating object and r is the radius of rotation. However, if we are given the rotational velocity in revolutions per minute, it can be convertible to angular velocity using the formula below: - $ω = 2π· RPM/60$ Multiply the rotational velocity by 2n since there are 2n radians in one revolution and divide by 60 seconds since 1 min = 60 seconds. ## Chapter 1.4: Simulated Gravity ### The Concept Gravity can be simulated into objects that lack natural gravity. Through linear acceleration and centrifugal force (rotation), gravity is mimicked upon objects and living organisms, particularly astronauts. Gravity can be artificially mimicked through the help of linear acceleration and rotation (centrifugal force and centripetal force). Let's talk about how these two contribute to artificial gravity: ## Centrifugal Force (Rotation) **Key Takeaway** Centrifugal force is the force that pulls an object away from the center of rotation. It is not a real force but it is a product of inertia acting on a mass upon a system. _**Critical Analysis**_ Have you ever wondered why water splashes when a car runs over a pool of water in the road? This happens because of the principle of centrifugal force. **Figure 15. A car running over a pool of water** The wheels are in motion, that's why they generate centrifugal force. At the moment that the water touches the wheels, the water molecules attach to the tires (adhesive forces) and the wheels swing them in a swirling pattern. Centrifugal force is what pushes the water away from the wheels. We can see in Figure 15 some visible water swirls brought by the car. ### How can it simulate gravity? It is not new in our senses that in one way or another, we might live in space someday sometime soon. If this happens, we will most likely live in a simulated habitat, but the lack of gravity is what prohibits us from experiencing earth-like gravity. So, these artificial environments need to rotate to simulate gravity through centrifugal force and it needs to achieve linear acceleration due to gravity 9.81 $m/s^2$. Centrifugal force can be mathematically expressed as follows: - $F = mrw^2$ ## Centripetal Force In the presence of centrifugal force upon a rotating object, there is also another force that counterparts it - the centripetal force. Centripetal force is the force that retains an object in its position in a rotating object by pulling it toward the center. _**Critical Analysis**_ Unlike most forces, centripetal force, on its own does not exist and it depends upon the context on how it's used. Here are the keynote purposes of centripetal force upon different contexts. 1. Planetary Orbit – The sun's gravity is what provides the centripetal force by pulling the planets toward it so that they will stay in place and orbit. 2. String Tension – A hanging pendulum, for example, experiences centripetal force because of the tension present in the string. Since tension is an upward force, the pendulum will eventually stop because the tension is pulling the ball toward its center of rotation. 3. Turning Car – The friction between the wheels and the road of a turning car provides the centripetal force needed in order to safely steer the car. Without friction, the car will almost move in a perfectly linear path no matter how you turn the pedal. If astronauts, someday, could possibly in artificial generated habitats, they can stay in place while their habitat is rotating with the help of centripetal force. Note that centripetal force is equal to centrifugal force which can be expressed as follows: - $F = mrw^2$ ## Linear Acceleration _**Critical Analysis**_ Due to the challenges of today's technology, scientists in the future may resort to linear acceleration in order to simulate gravity. Imagine that you are in a stationary spacecraft, along with your fellow astronauts. You won't feel the effect of gravity. That is, until the spacecraft moves at an acceleration rate of 9.81 $m/s^2$. That's right – the spacecraft would need to continuously move in a straight line to simulate gravity. This option also has a lot of disadvantages, including fuel efficiency and the awkward turbulence brought by the continuously moving spacecraft. Would you want to still simulate gravity, or you'd rather stay afloat in space? ## Chapter 1.5: Rotational Kinematics ### The Concept Rotational kinematics is almost the same with linear kinematics. Aside from major changes in variables, some formula changes heavily depend on angular displacement in which depends on the radial vector and the length of the arc in which the object is displaced. Imagine an object, preferably a person standing on a round-a-bout. As the round-a-bout spins, the object gets displaced from its original position depending upon the the length of the arc l in which the object is displaced and the radial vector r. This is the simplest description of angular displacement. To better understand it, look at the figure below: **Figure 16. Angular Displacement on a Round-a-Bout** Like what I have said earlier, there will be major variable changes to the original linear kinematics formulas but before that, we will start with the fundamental formula for rotational motion – the angular displacement. Given the radial vector (radius of the rotating object) r and the length of the arc of displacement l, the formula for angular displacement can be expressed as follows: - $θ = l/r$ The rest of the formulas shall follow. Angular displacement is positive when it is counterclockwise and negative if it's clockwise. Given the variables for angular velocity ω, angular acceleration α, and angular displacement as θ, the updated formulas for rotational motion are: - $ωf = ω₁ + at$ - $θ = ω₁t + 1/2 at^2$ - $θ = ωω/2α$ - $θ = (ω₁ + ω₂ +………ωn)/n$ Angular velocity can be expressed as the derivative of angular displacement with respect to time ω = $dθ/dt$ and angular acceleration can be expressed as the derivative of angular velocity with respect to time α = $dω/dt$ ## Tangential Acceleration and Velocity _**Key Takeaway**_ Both tangential acceleration and velocity talk about the linear change in motion of an object moving in a circular path. Since its linear, we will expect for a unit of m/s² and m/s. Since it is relatively like linear motion, the conversion of this into angular acceleration and velocity is already tackled in **formula [1.7]**.