Rotational Kinematics Physics Problems, Basic Introduction, Equations & Formulas

The Organic Chemistry Tutor

27 Oct 201719:33

EducationalLearning

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TLDRThis video script delves into the realm of rotational kinematics, presenting essential formulas for understanding the motion of objects in both linear and angular contexts. It begins by establishing the foundational equations for constant speed and acceleration, drawing parallels between linear and angular motion. The script then explores the relationship between linear and angular displacement, velocity, and acceleration, emphasizing the importance of radius in these calculations. Practical problems are solved throughout, such as calculating the number of revolutions a wheel makes given a constant angular speed, determining the linear distance traveled by a wheel, and finding the final angular and linear speeds of a disk accelerating from rest. The video concludes with a problem involving a wheel's acceleration and the calculation of its revolutions over a period of time. The script is an excellent educational resource for those looking to grasp the principles of rotational kinematics and apply them to real-world scenarios.

Takeaways

📐 **Linear vs. Angular Motion**: The video differentiates between linear (translational) and angular (rotational) motion, highlighting their respective equations.
⚙️ **Constant Speed Equation**: For constant speed (zero acceleration), displacement (d) is equal to velocity (v) times time (t), and angular displacement (θ) equals angular velocity (ω) times time (t).
⏱️ **Constant Acceleration Equations**: The final velocity (v or ω) can be found using the initial velocity plus acceleration times time for both linear and angular motion.
🔄 **Final Velocity and Acceleration**: The relationship between final velocity, initial velocity, and acceleration is expressed by the equations v² = u² + 2ad and ω² = ω₀² + 2αθ.
📉 **Displacement with Constant Acceleration**: Displacement can be calculated using the formula d = ut + 1/2at² for linear motion and θ = ω₀t + 1/2αt² for angular motion.
🔵 **Average Speed and Angular Speed**: The average speed or velocity is the sum of initial and final speeds divided by two times time, and similarly for angular motion.
🔗 **Linear and Angular Relationships**: Linear displacement equals angular displacement times the radius, and similarly, linear velocity equals angular velocity times the radius, and linear acceleration equals angular acceleration times the radius.
⭕ **Revolutions from Angular Speed**: To find the number of revolutions from a given angular speed and time, use the formula θ = ωt and convert radians to revolutions by dividing by 2π.
🚀 **Linear Distance from Angular Motion**: The linear distance traveled by a point on a rotating object can be found by multiplying the angular distance in radians by the radius.
🛑 **Angular Speed from Acceleration**: The final angular speed can be calculated from the initial angular speed, angular acceleration, and time using the equation ω = ω₀ + αt.
🔍 **Revolutions from Angular Acceleration**: The total angle in radians turned by an object can be found using θ = ω₀t + 1/2αt², and then converted to revolutions.
📐 **Linear Speed from Angular Speed**: The linear speed at the edge of a rotating disk is found by multiplying the angular speed by the radius of the disk.
🔧 **Calculating Angular Acceleration**: Given the linear acceleration and radius, the angular acceleration can be found by dividing the linear acceleration by the radius.
🔢 **Average Angular Acceleration**: The average angular acceleration is calculated by the change in angular speed divided by the time over which the change occurred.

Q & A

What is the first formula to know for an object moving with constant speed in linear motion?
-The first formula is d = vt, where d represents displacement or distance, v is the linear speed or velocity, and t is time.
What is the corresponding formula for an object moving with constant speed in angular motion?
-The corresponding formula for angular motion is θ = ωt, where θ is the angular displacement, ω is the angular speed or angular velocity, and t is time.
If an object has a constant acceleration, what is the formula that relates the final velocity to the initial velocity, acceleration, and time?
-The formula relating the final velocity (v_final) to the initial velocity (v_initial), acceleration (a), and time (t) is v_final = v_initial + at.
What is the angular version of the formula for an object with constant acceleration in linear motion?
-The angular version of the formula is ω_final = ω_initial + αt, where ω_final is the final angular velocity, ω_initial is the initial angular velocity, α is the angular acceleration, and t is time.
How is the final squared velocity related to the initial squared velocity, acceleration, and displacement in linear motion?
-The relationship is given by the formula v_final^2 = v_initial^2 + 2ad, where v_final is the final velocity, v_initial is the initial velocity, a is the acceleration, and d is the displacement.
What is the formula that relates linear displacement to angular displacement and the radius of a rotating object?
-The formula is linear displacement = angular displacement × radius, which is similar to the trigonometric formula for arc length as the angle in radians times the radius.
If a wheel spins at a constant angular speed of 24 radians per second, how many revolutions will it go through in 5 minutes?
-The wheel will go through 1146 revolutions in 5 minutes, calculated by multiplying the angular speed (24 radians/second) by the time in seconds (300 seconds) and then dividing by 2π radians per revolution.
How can you find the linear distance traveled by a point on the ground in 5 minutes, given the wheel's radius and angular speed?
-You can find the linear distance by multiplying the angular distance in radians (calculated from the wheel's angular speed and time) by the radius of the wheel.
What is the final angular speed of a disk that accelerates from rest at a constant rate of 2.5 radians per second squared for 18 seconds?
-The final angular speed of the disk is 45 radians per second, found by using the formula ω_final = ω_initial + αt, with ω_initial being 0, α being 2.5 rad/s², and t being 18 seconds.
How many revolutions will a disk go through if it accelerates from rest at a rate of 2.5 radians per second squared for 18 seconds?
-The disk will go through approximately 64.46 revolutions, calculated by using the formula θ = ω_initialt + 0.5αt², and then converting the angular distance θ from radians to revolutions.
What is the linear speed of a point at the edge of a disk after 18 seconds, given the final angular speed and a radius of 0.5 meters?
-The linear speed of the point is 22.5 meters per second, calculated by multiplying the final angular speed (45 radians per second) by the radius of the disk (0.5 meters).
If a wheel with a diameter of 80 centimeters accelerates from 30 radians per second to 80 radians per second, and the linear acceleration at the edge of the wheel is 15 meters per second squared, how many revolutions will the wheel go through during this period?
-The wheel will go through approximately 11.67 revolutions, calculated by first finding the angular acceleration from the given linear acceleration and then using the formula θ = (ω_final² - ω_initial²) / (2α) to find the angular distance in radians, which is then converted to revolutions.

Outlines

00:00

📐 Introduction to Rotational Kinematics

The video introduces the topic of solving problems related to rotational kinematics. It provides a list of essential formulas for linear and angular motion, emphasizing the similarities between the two. The linear equations are for translational motion, and the angular equations are for rotational motion. Key formulas include constant speed motion (d = vt and theta = omega * t), constant acceleration motion (v_final = v_initial + at and omega_final = omega_initial + alpha * t), and equations for final velocity in terms of initial velocity and acceleration (v_final^2 = v_initial^2 + 2ad and omega_final^2 = omega_initial^2 + 2 * alpha * theta). The video also discusses additional formulas relating linear and angular displacement, velocity, and acceleration through the radius of rotation.

05:01

🔄 Problem Solving: Constant Angular Speed

The first problem involves a wheel spinning at a constant angular speed of 24 radians per second for 5 minutes. The main formula used is theta = omega * t, where omega is given and t is converted from minutes to seconds. The angular distance in radians is calculated, and then converted to revolutions using the fact that one revolution equals 2 pi radians. The wheel is found to make 1146 revolutions. For part B, the linear distance traveled along the ground is calculated using the arc length formula (linear distance = angular displacement * radius), assuming a radius of 2 meters, resulting in a distance of 14,400 meters.

10:01

🚀 Accelerating Disk: Angular Speed and Revolutions

The second problem features a disk accelerating from rest at a constant rate of 2.5 radians per second squared for 18 seconds. The formula omega_final = omega_initial + alpha * t is used to find the final angular speed, which is calculated to be 45 radians per second. To find the number of revolutions, the formula theta = omega_initial * t + 0.5 * alpha * t^2 is applied, yielding 405 radians, which converts to approximately 64.46 revolutions.

15:01

⚙️ Calculations for Linear Speed and Acceleration

The third problem involves calculating the linear speed of a point at the edge of a disk after 18 seconds, given the final angular speed of 45 radians per second and a radius of 0.5 meters. The linear speed is found by multiplying the angular speed by the radius, resulting in 22.5 meters per second. For a disk with a diameter of 60 centimeters accelerating from 20 to 40 radians per second in 5 seconds, the average angular speed is used to calculate the angular distance in radians, which converts to 23.87 revolutions. The average angular acceleration is determined using the change in angular speed over time, resulting in 4 radians per second squared. The average linear acceleration is then found by multiplying the angular acceleration by the radius, yielding 1.2 meters per second squared.

🛤️ Wheel Revolutions with Given Linear Acceleration

The final problem concerns a wheel with a diameter of 80 centimeters accelerating from 30 to 80 radians per second with a given linear acceleration of 15 meters per second squared. The radius is calculated as 0.4 meters. Using the linear acceleration to find the angular acceleration (alpha = linear acceleration / radius), the value of alpha is determined to be 37.5 radians per second squared. The formula omega_final^2 = omega_initial^2 + 2 * alpha * theta is then used to solve for theta, the angular distance the wheel goes through, which is found to be 73.33 radians or approximately 11.67 revolutions.

Mindmap

Keywords

💡Rotational Kinematics

Rotational kinematics is the study of the motion of rotating objects. It is a branch of physics that deals with angular displacement, velocity, and acceleration. In the video, it is the central theme as the script discusses how to solve problems associated with it using various formulas and equations.

💡Constant Speed

Constant speed refers to a situation where an object moves at a uniform rate with zero acceleration. In the context of the video, it is used to describe linear and angular motion where the displacement (d) or angular displacement (theta) is directly proportional to time (t), as in the formula d = vt or theta = omega * t.

💡Angular Displacement

Angular displacement is the angle through which a rotating object moves from its initial position. It is measured in radians and is used to describe the rotation in the context of rotational kinematics. The video script uses this term to explain how it is calculated using the formula theta = omega * t, where omega is the angular velocity.

💡Linear and Angular Velocity

Linear velocity (v) is the rate at which an object moves in a straight line, while angular velocity (omega) is the rate of rotation around an axis. The video script draws parallels between these two concepts, noting that they are analogous in their respective motion types, with equations like v = d/t and omega = theta/t.

💡Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a constant rate over time. The video script discusses how to calculate final velocity (v or omega) using the initial velocity, acceleration (a or alpha), and time with the formula v_final = v_initial + a*t or omega_final = omega_initial + alpha*t.

💡Arc Length

Arc length is the distance along the curved path of a rotation. In the video, it is related to linear displacement when discussing the motion of a wheel or a rotating object. The script mentions that linear displacement is equal to the angular displacement times the radius, which is similar to the trigonometric concept of arc length being the angle in radians times the radius.

💡Revolutions

Revolutions refer to the number of complete cycles or turns that a rotating object makes. The video script uses this term to convert angular displacement from radians to a more understandable unit of measurement, especially when calculating the rotation of a wheel or disc over a period of time.

💡Angular Acceleration

Angular acceleration (alpha) is the rate of change of angular velocity. It is analogous to linear acceleration (a) in translational motion. The video script explains how to calculate the final angular speed and the number of revolutions using angular acceleration in equations like omega_final^2 = omega_initial^2 + 2*alpha*theta.

💡Average Angular Speed

Average angular speed is the mean of the initial and final angular speeds over a period of time. It is used in the video to calculate the angular distance covered during a certain time interval, especially when the acceleration is not constant. The script uses the formula theta = (1/2) * (omega_initial + omega_final) * t.

💡Linear Acceleration

Linear acceleration is the rate at which an object's linear velocity changes over time. In the video, it is derived from angular acceleration by multiplying it with the radius of the rotating object. The script uses this concept to find the average linear acceleration of a disk using the formula linear acceleration = alpha * r.

💡Radius

The radius is the distance from the center of a circle or rotation to its edge. It is a crucial component in the equations relating linear and angular quantities, such as linear velocity being equal to angular velocity times the radius. The video script uses the radius to calculate linear displacement, linear speed, and linear acceleration from their angular counterparts.

Highlights

The video discusses solving problems associated with rotational kinematics.

A list of essential formulas for linear and angular equations is provided.

Constant speed motion is described by the equation d = vt, where d is displacement or distance, and v is velocity.

Angular displacement is calculated using theta = omega * t, with omega being angular velocity.

For motion with constant acceleration, the final velocity is given by v_final = v_initial + a*t.

The angular equivalent for constant acceleration is omega_final = omega_initial + alpha*t.

The relationship between final squared velocity and initial velocity is shown as v_final^2 = v_initial^2 + 2ad.

The angular version of the above equation is omega_final^2 = omega_initial^2 + 2*alpha*theta.

Displacement can be calculated using the average speed or velocity with the equation d = (v_initial + v_final) / 2 * t.

The relationship between linear displacement and angular displacement is given by d = theta * r, where r is the radius.

Linear velocity is equal to angular velocity times the radius, and linear acceleration equals angular acceleration times the radius.

A problem-solving example is provided where a wheel spins at a constant angular speed of 24 radians per second for 5 minutes.

The wheel's angular distance in radians and the number of revolutions completed in 5 minutes are calculated.

The linear distance traveled by the wheel in 5 minutes is determined using the arc length formula.

Another example calculates the final angular speed of a disk accelerating at 2.5 radians per second squared for 18 seconds.

The number of revolutions and the linear speed at the edge of the disk after 18 seconds are determined.

A third problem involves a disc accelerating from 20 to 40 radians per second in 5 seconds, with calculations for revolutions, angular acceleration, and linear acceleration.

The final problem calculates the number of revolutions a wheel goes through when accelerating from 30 to 80 radians per second with a given linear acceleration.

Transcripts

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Takeaways

Q & A

What is the first formula to know for an object moving with constant speed in linear motion?

What is the corresponding formula for an object moving with constant speed in angular motion?

If an object has a constant acceleration, what is the formula that relates the final velocity to the initial velocity, acceleration, and time?

What is the angular version of the formula for an object with constant acceleration in linear motion?

How is the final squared velocity related to the initial squared velocity, acceleration, and displacement in linear motion?

What is the formula that relates linear displacement to angular displacement and the radius of a rotating object?

If a wheel spins at a constant angular speed of 24 radians per second, how many revolutions will it go through in 5 minutes?

How can you find the linear distance traveled by a point on the ground in 5 minutes, given the wheel's radius and angular speed?

What is the final angular speed of a disk that accelerates from rest at a constant rate of 2.5 radians per second squared for 18 seconds?

How many revolutions will a disk go through if it accelerates from rest at a rate of 2.5 radians per second squared for 18 seconds?

What is the linear speed of a point at the edge of a disk after 18 seconds, given the final angular speed and a radius of 0.5 meters?

If a wheel with a diameter of 80 centimeters accelerates from 30 radians per second to 80 radians per second, and the linear acceleration at the edge of the wheel is 15 meters per second squared, how many revolutions will the wheel go through during this period?