6 min read•march 22, 2023
Peter Apps
Kashvi Panjolia
Peter Apps
Kashvi Panjolia
Enduring Understanding 4.D
A exerted on a system by other objects or systems will change the of the system.
Essential Knowledge 4.D.1
, , , and are vectors and can be characterized as positive or negative depending on whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis.
Essential Knowledge 4.D.2
The of a system may change due to interactions with other objects or systems.
Essential Knowledge 4.D.3
The change in is given by the product of the average and the time interval during which the is exerted.
is the rotational equivalent of linear momentum. It is represented by the equation L=I𝜔 where L is the , I is the of the object, and 𝜔 is the of the object. In addition, the of an object moving in a straight line relative to a fixed point can be found by multiplying its linear momentum (p) by the perpendicular distance (r) to the point as shown in the picture.
An object that is moving in a straight line can still have if the is calculated relative to a fixed point. This is because if you are standing at that fixed point and looking at an object moving past you, you will still have to turn your head to watch that object, which is the source of the . For example, if you are standing at a train station and watch a train go by, you will first turn your head to the right (or left) to watch the train come in, then look straight ahead as the train stops at the station, and finally, turn your head to the left (or right) to watch the train leave. In this situation, the train has an about a fixed point, which is your position on the platform. This can be calculated using L=mvr, since you will usually not be given an angle value.
🎥Watch: Minutephysics - What is Angular Momentum?
Just like a force acting on an object causes a change in linear momentum, a causes a change in the of the object. Also, the change in linear momentum (impulse) of an object can be calculated by multiplying the net force exerted on the object by the amount of time the force was exerted. Similarly, the change in of an object can be calculated by the exerted on the object multiplied by the time interval for which the was exerted. The units for are kilogram meters squared per second or kgm^2/s.
It is also important to note that the net of a system is the vector sum of the of each object in the system. This is because is a and follows the rules of vector addition. The total of the system is the sum of all the individual angular momenta of the objects in the system, and the direction of the total is determined by the directions of the individual angular momenta.
The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot.
A student will slide a disk of mass:
toward the rod with velocity v0 perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot.
The student wants the rod-disk system to end up with as much as possible.
Briefly explain your reasoning without manipulating equations.
CORRECT ANSWER: To the right of C
REASONING: To make the largest , the rod needs to get the greatest possible (L=I𝜔). To get the largest , the rod must be given the greatest possible . Since depends on the and the (𝜏=Iα), and the is calculated using I=cMR^2 (where c is a constant depending on the object), hitting the rod at the greatest radius will produce the greatest . Because the pivot is on the left side of C, we want to hit on the right side of C to maximize the radius so the increases, causing the to increase, which will increase the .
ALTERNATE EXPLANATION: To create the largest , the rod needs to get the greatest possible (L=I𝜔). To do this, we want the disk to transfer the greatest amount of momentum to the rod. Since is conserved in the rod-disk system (no outside torques), this means that we want the initial of the disk to be as large as possible. The of the disk is equal to L=mvr where r is the distance from the pivot because the disk is moving in a straight-line motion relative to the pivot point of the rod.
To make:
the largest it can possibly be, it needs to hit at the greatest distance away from the pivot to maximize the value of r. The greatest distance away from the pivot is on the right side of C.
(This explanation uses the which will be covered in the next section.)
🎥Watch: AP Physics 1 - Unit 7 Streams
1. A spinning disk has a mass of 0.5 kg and a radius of 0.2 m. The disk is spinning at an of 2 rad/s. What is the of the disk? (I = 1/2MR^2 for the disk)
A) 0.02 kgm^2/s B) 0.2 kgm^2/s C) 0.4 kgm^2/s D) 0.8 kgm^2/s
The solution is: L = Iω = (1/2) mr^2ω = (1/2)0.5(0.2^2)2 = 0.02 kgm^2/s
The answer is A) 0.02 kgm^2/s
2. A 0.4 kg ball is attached to a string of length 0.5 m and is rotating in a circle with an of 3 rad/s. What is the of the ball? (I= 2/5MR^2 for the ball)
A) 0.3 kgm^2/s B) 0.12 kgm^2/s C) 0.9 kgm^2/s D) 1.2 kgm^2/s
The solution is:
L = Iω = (2/5) mr^2ω = (2/5)0.4(0.5^2)3 = 0.12 kgm^2/s The answer is:
A) 0.12 kgm^2/s
3. A flywheel (disk) with a mass of 8 kg and a radius of 0.3 m is spinning at an of 4 rad/s. What is the of the flywheel? (I = 1/2MR^2 for the flywheel)
A) 1.44 kgm^2/s B) 2.4 kgm^2/s C) 4.8 kgm^2/s D) 9.6 kgm^2/s
The solution is:
L = Iω = (1/2) mr^2ω = (1/2)8(0.3^2)4 = 1.44 kgm^2/s The answer is
A) 1.44 kgm^2/s
Angular Acceleration
: Angular acceleration refers to the rate at which an object's angular velocity changes over time. It measures how quickly an object is speeding up or slowing down its rotation.Angular momentum
: Angular momentum refers to the rotational equivalent of linear momentum. It describes how fast an object rotates around an axis and depends on its mass distribution and rotational speed.Angular Speed
: Angular speed refers to how fast an object is rotating or spinning around a fixed axis. It is measured in radians per second (rad/s).Angular Velocity
: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).Law of Conservation of Momentum
: The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In other words, the total momentum before an event is equal to the total momentum after the event.Moment of Inertia
: Moment of inertia measures an object's resistance to changes in its rotational motion. It depends on both the mass distribution and the axis of rotation.Net Torque
: Net torque refers to the sum of all torques acting on an object. It determines the rate at which an object's rotational motion changes and can cause objects to accelerate or decelerate their rotation.Rotational Dynamics
: Rotational dynamics refers to the study of how objects rotate around an axis under the influence of torques (rotational forces). It involves concepts such as angular velocity, moment of inertia, and torque.Torque
: Torque refers to the measure of how effectively a force can cause an object to rotate around a fixed axis. It depends on both the magnitude and direction of the applied force.Vector Quantity
: A vector quantity is a physical quantity that has both magnitude and direction. It can be represented by an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.6 min read•march 22, 2023
Peter Apps
Kashvi Panjolia
Peter Apps
Kashvi Panjolia
Enduring Understanding 4.D
A exerted on a system by other objects or systems will change the of the system.
Essential Knowledge 4.D.1
, , , and are vectors and can be characterized as positive or negative depending on whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis.
Essential Knowledge 4.D.2
The of a system may change due to interactions with other objects or systems.
Essential Knowledge 4.D.3
The change in is given by the product of the average and the time interval during which the is exerted.
is the rotational equivalent of linear momentum. It is represented by the equation L=I𝜔 where L is the , I is the of the object, and 𝜔 is the of the object. In addition, the of an object moving in a straight line relative to a fixed point can be found by multiplying its linear momentum (p) by the perpendicular distance (r) to the point as shown in the picture.
An object that is moving in a straight line can still have if the is calculated relative to a fixed point. This is because if you are standing at that fixed point and looking at an object moving past you, you will still have to turn your head to watch that object, which is the source of the . For example, if you are standing at a train station and watch a train go by, you will first turn your head to the right (or left) to watch the train come in, then look straight ahead as the train stops at the station, and finally, turn your head to the left (or right) to watch the train leave. In this situation, the train has an about a fixed point, which is your position on the platform. This can be calculated using L=mvr, since you will usually not be given an angle value.
🎥Watch: Minutephysics - What is Angular Momentum?
Just like a force acting on an object causes a change in linear momentum, a causes a change in the of the object. Also, the change in linear momentum (impulse) of an object can be calculated by multiplying the net force exerted on the object by the amount of time the force was exerted. Similarly, the change in of an object can be calculated by the exerted on the object multiplied by the time interval for which the was exerted. The units for are kilogram meters squared per second or kgm^2/s.
It is also important to note that the net of a system is the vector sum of the of each object in the system. This is because is a and follows the rules of vector addition. The total of the system is the sum of all the individual angular momenta of the objects in the system, and the direction of the total is determined by the directions of the individual angular momenta.
The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot.
A student will slide a disk of mass:
toward the rod with velocity v0 perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot.
The student wants the rod-disk system to end up with as much as possible.
Briefly explain your reasoning without manipulating equations.
CORRECT ANSWER: To the right of C
REASONING: To make the largest , the rod needs to get the greatest possible (L=I𝜔). To get the largest , the rod must be given the greatest possible . Since depends on the and the (𝜏=Iα), and the is calculated using I=cMR^2 (where c is a constant depending on the object), hitting the rod at the greatest radius will produce the greatest . Because the pivot is on the left side of C, we want to hit on the right side of C to maximize the radius so the increases, causing the to increase, which will increase the .
ALTERNATE EXPLANATION: To create the largest , the rod needs to get the greatest possible (L=I𝜔). To do this, we want the disk to transfer the greatest amount of momentum to the rod. Since is conserved in the rod-disk system (no outside torques), this means that we want the initial of the disk to be as large as possible. The of the disk is equal to L=mvr where r is the distance from the pivot because the disk is moving in a straight-line motion relative to the pivot point of the rod.
To make:
the largest it can possibly be, it needs to hit at the greatest distance away from the pivot to maximize the value of r. The greatest distance away from the pivot is on the right side of C.
(This explanation uses the which will be covered in the next section.)
🎥Watch: AP Physics 1 - Unit 7 Streams
1. A spinning disk has a mass of 0.5 kg and a radius of 0.2 m. The disk is spinning at an of 2 rad/s. What is the of the disk? (I = 1/2MR^2 for the disk)
A) 0.02 kgm^2/s B) 0.2 kgm^2/s C) 0.4 kgm^2/s D) 0.8 kgm^2/s
The solution is: L = Iω = (1/2) mr^2ω = (1/2)0.5(0.2^2)2 = 0.02 kgm^2/s
The answer is A) 0.02 kgm^2/s
2. A 0.4 kg ball is attached to a string of length 0.5 m and is rotating in a circle with an of 3 rad/s. What is the of the ball? (I= 2/5MR^2 for the ball)
A) 0.3 kgm^2/s B) 0.12 kgm^2/s C) 0.9 kgm^2/s D) 1.2 kgm^2/s
The solution is:
L = Iω = (2/5) mr^2ω = (2/5)0.4(0.5^2)3 = 0.12 kgm^2/s The answer is:
A) 0.12 kgm^2/s
3. A flywheel (disk) with a mass of 8 kg and a radius of 0.3 m is spinning at an of 4 rad/s. What is the of the flywheel? (I = 1/2MR^2 for the flywheel)
A) 1.44 kgm^2/s B) 2.4 kgm^2/s C) 4.8 kgm^2/s D) 9.6 kgm^2/s
The solution is:
L = Iω = (1/2) mr^2ω = (1/2)8(0.3^2)4 = 1.44 kgm^2/s The answer is
A) 1.44 kgm^2/s
Angular Acceleration
: Angular acceleration refers to the rate at which an object's angular velocity changes over time. It measures how quickly an object is speeding up or slowing down its rotation.Angular momentum
: Angular momentum refers to the rotational equivalent of linear momentum. It describes how fast an object rotates around an axis and depends on its mass distribution and rotational speed.Angular Speed
: Angular speed refers to how fast an object is rotating or spinning around a fixed axis. It is measured in radians per second (rad/s).Angular Velocity
: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).Law of Conservation of Momentum
: The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In other words, the total momentum before an event is equal to the total momentum after the event.Moment of Inertia
: Moment of inertia measures an object's resistance to changes in its rotational motion. It depends on both the mass distribution and the axis of rotation.Net Torque
: Net torque refers to the sum of all torques acting on an object. It determines the rate at which an object's rotational motion changes and can cause objects to accelerate or decelerate their rotation.Rotational Dynamics
: Rotational dynamics refers to the study of how objects rotate around an axis under the influence of torques (rotational forces). It involves concepts such as angular velocity, moment of inertia, and torque.Torque
: Torque refers to the measure of how effectively a force can cause an object to rotate around a fixed axis. It depends on both the magnitude and direction of the applied force.Vector Quantity
: A vector quantity is a physical quantity that has both magnitude and direction. It can be represented by an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.© 2024 Fiveable Inc. All rights reserved.
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