Physics
Torque Vector
Torque vector refers to a quantity that represents the rotational effect of a force. It has both magnitude and direction, and its direction is perpendicular to the plane formed by the force and the point of rotation. In physics, torque vectors are used to describe the rotational motion of objects and are essential in understanding the behavior of rotating systems.
Written by Perlego with AI-assistance
Related key terms
1 of 5
5 Key excerpts on "Torque Vector"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The large counterweight on the right side (short end) of this tall tower crane ensures its boom remains balanced on its mast while lifting heavy loads. It is not equal weights on both sides of the tower that keep it in equilibrium, but equal torques. Torque is the rotational analog of force, and is an important topic of this chapter. 9 | Rotational Dynamics Chapter | 9 LEARNING OBJECTIVES After reading this module, you should be able to... 9.1 | Distinguish between torque and force. 9.2 | Analyze rigid objects in equilibrium. 9.3 | Determine the center of gravity of rigid objects. 9.4 | Analyze rotational dynamics using moments of inertia. 9.5 | Apply the relation between rotational work and energy. 9.6 | Solve problems using the conservation of angular momentum. 9.1 | The Action of Forces and Torques on Rigid Objects The mass of most rigid objects, such as a propeller or a wheel, is spread out and not con- centrated at a single point. These objects can move in a number of ways. Figure 9.1a il- lustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9.1b. We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration. A net external force causes linear motion to change, but what causes rotational motion to change? For example, something causes the rotational velocity of a speedboat’s propeller to change when the boat accelerates.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 9 Rotational dynamics LEARNING OBJECTIVES After reading this module, you should be able to: 9.1 distinguish between torque and force 9.2 analyse rigid objects in equilibrium 9.3 determine the centre of gravity of rigid objects 9.4 analyse rotational dynamics using moments of inertia 9.5 apply the relation between rotational work and energy 9.6 solve problems using the conservation of angular momentum. INTRODUCTION The large counterweight on the right side (short end) of this tall tower crane ensures its boom remains balanced on its mast while lifting heavy loads. It is not equal weights on both sides of the tower that keep it in equilibrium, but equal torques. Torque is the rotational analog of force, and is an important topic of this chapter. Source: Mr. Green / Shutterstock 9.1 The action of forces and torques on rigid objects LEARNING OBJECTIVE 9.1 Distinguish between torque and force. The mass of most rigid objects, such as a propeller or a wheel, is spread out and not concentrated at a single point. These objects can move in a number of ways. Figure 9.1a illustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in figure 9.1b. FIGURE 9.1 Examples of (a) translational motion and (b) combined translational and rotational motions Translation ( ) a Combined translation and rotation ( ) b We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
In summary, the torque depends on the magnitude of the force, on the point where the force is applied relative to the axis of rotation (the hinge in Interac- tive Figure 9.2), and on the direction of the force. For simplicity, we deal with situations in which the force lies in a plane that is per- pendicular to the axis of rotation. In Figure 9.3, for instance, the axis is perpendicular to the page and the force lies in the plane of the paper. The drawing shows the line of action and the lever arm of the force, two concepts that are important in the definition of torque. The line of action is an extended line drawn colinear with the force. The lever arm is the distance ℓ between the line of action and the axis of rotation, measured on a line that is perpendicular to both. The torque is represented by the symbol τ (Greek letter tau), and its magnitude is defined as the magnitude of the force times the lever arm: DEFINITION OF TORQUE Magnitude of torque = (Magnitude of the force) × (Lever arm) = Fℓ (9.1) Direction: The torque τ is positive when the force tends to produce a counterclockwise rotation about the axis, and negative when the force tends to produce a clockwise rotation. SI Unit of Torque: newton · meter (N · m) Equation 9.1 indicates that forces of the same magnitude can produce different torques, depending on the value of the lever arm, and Example 1 illustrates this important feature. F F F ( ) a Hinge (axis of rotation) ( ) b ( ) c Door INTERACTIVE FIGURE 9.2 With a force of a given magnitude, a door is easier to open by (a) pushing at the outer edge than by (b) pushing closer to the axis of rotation (the hinge). (c) Pushing into the hinge makes it difficult to open the door. EX AMPLE 1 Different Lever Arms, Different Torques In Figure 9.3 a force (magnitude = 55 N) is applied to a door. However, the lever arms are different in the three parts of the drawing: (a) ℓ = 0.80 m, (b) ℓ = 0.60 m, and (c) ℓ = 0 m.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
How effective a force is in closing the monitor depends on its magnitude, direction, and point of application. I N T E R A C T I V E F E A T U R E 238 | Chapter 9 We have resolved the force into two components: θ F cos , which is parallel to r , and θ F sin , which is perpendicular to r . The component θ F cos pulls directly away from the axis; intuitively we know that pulling away from the hinges will not cause the door to rotate. The component θ F sin , on the other hand, causes the door to rotate counterclock- wise. Thus, it is the component of F that is perpendicular to r that causes rotation. The quantity called torque is a quantitative measure of the extent to which a force will effect a rotation of an object about a particular axis, and is defined as follows: The SI unit of torque is newtons times meters, ⋅ N m. [Recall from Chapter 6 that ⋅ = 1 N m 1 J (joule), but we do not report torque in joules because torque is not a measure of energy.] The ± symbol indicates that the torque can be positive or negative, which will be discussed shortly. Figure 9.1.3 helps illustrate the dependence of the torque τ on the angle θ . Part (a) is a graph of θ sin versus θ . In part (b), three forces of the same magnitude, but different directions, are applied to the end of a wrench in an attempt to loosen a hex nut. The point of application is the same for all three forces, so r in Equation 9.1.1 is the same. Axis of rotation r Point of application of F F F cos θ θ F sin θ Figure 9.1.2 The vector r extends from the axis to the point of application of F . The component of F that is parallel to r causes no rotation. The component of F that is perpendicular to r causes rotation. The distance r is not necessarily a radius, but if the object is rotating about the axis, r is the radius of the circular path traveled by the point of application of the force.
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Note that this magnitude is consistent with the cross-product magnitude. Moreover, the direction of torque is consistent with the direction given by the right-hand rule, assuming r is the first vector and F the second. Thus, torque can be written as a cross product: = r × F (13.7) Similarly, the (orbital) rotational momentum of a point particle with momen- tum p located at a position r with respect to the point of interest can also be written as a cross product: = r × p (13.8) It’s worth reiterating that the magnitude of the torque is given by = rF sin , and this can be written as either = rF ⟂ or = r ⟂ F. In one situation it might be easier to find the component of F perpendicular to r, while in another it could be easier to find the component of r perpendicular to F. A similar idea holds true for the magnitude of the (orbital) angular momentum: = rp sin = rp ⟂ = r ⟂ p. 13.6.2. Activity: Calculating Torque a. The diagram below shows a meterstick (L = 1 m, m = 50 g) that is in the process of tipping over. We wish to determine the torque about the point in contact with the ground due to the gravitational force F g , which, as always, acts through the center of mass. To begin, draw in the vectors r and F g on this diagram and show the angle between them. Use the right-hand rule to determine the direction of r × F g . 30° b. Sketch the component r ⟂ on the diagram above and calculate its value. UNIT 13: THE VECTOR NATURE OF ROTATIONAL MOTION 443 c. Sketch the component F g⟂ on the diagram above and calculate its value. d. Calculate the magnitude of the torque as rF g⟂ , r ⟂ F g , and rF g sin and show that you get the same result in each case. While there are multiple ways to calculate torque, you should get the same answer regardless of which method you use (assuming you do it correctly). You should choose the method that’s easiest in a given situation. 13.6.3. Activity: Rotational Momentum of a Moving Object a.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
