![Page 1: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/1.jpg)
Chapter 10
Dynamics of Rotational
Motion
![Page 2: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/2.jpg)
Torque
• Definition:• A force can be defined as something
that causes linear motion to change.• A torque can be similarly defined as
something that causes a change in rotational motion.
![Page 3: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/3.jpg)
• The torque applied to a rigid body depends on the magnitude of the applied force, the distance away from the point of rotation ( lever arm), and the direction of the applied force.
• We can express the torque as a vector cross product.
![Page 4: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/4.jpg)
Example
• The Achilles tendon of a person is exerting a force of magnitude 720 N on the heel at a point that is located 3.6 x 10-2 m away from the point of rotation.
• Determine the torque about the ankle (point of rotation).
• Assume the force is perpendicular to the radial arm.
![Page 5: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/5.jpg)
Picture of foot and Achilles tendon
3.6 x 10 -2 m
F
55o
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Solution
• The magnitude of the torque is:
![Page 7: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/7.jpg)
Torque and Angular Acceleration
• Consider a point on a rigid body.
• If a force is applied we can determine the tangential acceleration by Newton’s second law.
![Page 8: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/8.jpg)
• We can express this in terms of the angular acceleration by the following relation.
• The force on the particle can now be written as:
![Page 9: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/9.jpg)
• If we apply the definition of torque to our previous relation we get the following:
• We can use the vector triple cross product identity to rewrite this expression.
![Page 10: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/10.jpg)
Vector Triple Cross Product
A B C B A C C A B
![Page 11: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/11.jpg)
• If we apply this identity to our product we get the following:
• However, r and are perpendicular; therefore,
![Page 12: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/12.jpg)
• The torque now becomes:
• We can rewrite this in terms of the moment of inertia.
![Page 13: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/13.jpg)
Rigid-Body Rotations About a Moving Axis
• Consider the motion of a wheel on an automobile as it moves along the road.
• Since the wheel is in motion it possesses kinetic energy of motion.
• However, since the wheel is both translating along the road and rotating, the kinetic energy of the wheel is shared between these two motions.
![Page 14: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/14.jpg)
• The kinetic energy of a rotating body about a moving axis is given by:
![Page 15: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/15.jpg)
Example
• Determine the kinetic energy of a solid cylinder of radius R and mass M if it rolls without slipping.
![Page 16: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/16.jpg)
Solution
• If an object rolls without slipping the speed of the center of mass is related to the angular speed by the following:
![Page 17: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/17.jpg)
• The kinetic energy is given by:
• The moment of inertia for a solid cylinder is:
![Page 18: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/18.jpg)
• The kinetic energy then becomes:
![Page 19: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/19.jpg)
Work and Power for Rotations
• The work done rotating a rigid object through and infinitesimal angle about some axis is given by the following:
![Page 20: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/20.jpg)
• The total work done rotating the object from some initial angle to some final angle is:
![Page 21: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/21.jpg)
• If we examine the integrand we see that it can be rewritten as:
![Page 22: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/22.jpg)
• We integrate to get the work done.
![Page 23: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/23.jpg)
Power
• The power of a rotating body can be obtained by differentiating the work.
![Page 24: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/24.jpg)
• If we differentiate we get:
![Page 25: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/25.jpg)
Angular Momentum
• We define the linear moment of a system as:
• We saw that Newton’s second law can be written in terms of the linear momentum.
![Page 26: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/26.jpg)
• We can define momentum for rotations as well.
• We define the angular momentum of a system as:
• The value of the angular momentum depends upon the choice of origin.
• The units for angular momentum are:
2 /kg m s
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Newton’s Second Law
• Suppose an object is subjected to a net torque.
• How does this affect the angular momentum?
![Page 28: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/28.jpg)
• Suppose the angular momentum changes with time.
• We can write the following:
![Page 29: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/29.jpg)
• Now suppose the mass is constant.
• When we differentiate we get:
![Page 30: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/30.jpg)
• The first term is equal to zero due to the properties of the cross product.
• Therefore, the change in angular momentum is:
![Page 31: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/31.jpg)
Thus, we have Newton’s second law for rotations.
![Page 32: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/32.jpg)
• The angular momentum around a symmetry axis can be expressed in terms of the angular velocity.
![Page 33: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/33.jpg)
• If we take the time derivative of the angular momentum, once again we arrive at Newton’s second law.
I
![Page 34: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/34.jpg)
• A woman with mass 50-kg is standing on the rim of a large disk (a carousel) that is rotating at 0.50 rev/s about an axis through it center.
• The disk has mass 110-kg and a radius 4.0m.
• Calculate the magnitude of the total angular momentum of the woman plus disk system.
• Treat the woman as a point particle.
Example
![Page 35: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/35.jpg)
Solution
• The angular momentum of the system is:
• The magnitude of the angular momentum of the system is:
![Page 36: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/36.jpg)
• The magnitude of the linear momentum of the woman is:
• The magnitude of the angular momentum of the woman is:
![Page 37: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/37.jpg)
• The magnitude of the angular momentum of the disk is:
![Page 38: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/38.jpg)
• The magnitude of the total angular momentum of the system is:
![Page 39: Chapter 10 Dynamics of Rotational Motion. Torque Definition: A force can be defined as something that causes linear motion to change. A torque can be](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649d215503460f949f5df3/html5/thumbnails/39.jpg)
The Conservation of Angular Momentum
• Consider a system with no net torque acting upon it.
• The equation above implies that if the net torque on a system is zero, then the angular momentum of the system will be constant.