- Use the dot product to determine the work done on a moving object by a given force.
- Use the cross product to determine the force exerted on a moving charge by a magnetic field.
- Use the cross product to determine the torque exerted by a force on a object.
Vectors can be used by air-traffic controllers when tracking planes, by meteorologists when describing wind conditions, and by computer programmers when they are designing virtual worlds. In this section, we will present three applications of vectors which are commonly used in the study of physics: work, torque, and magnetic force.
In physics, the term work is used to describe energy that is added to or removed from an object or system when a force is applied to it. From experiment, it has been determined that work is maximized when the applied force is parallel to the motion of the object and that no work is done when the force is applied perpendicular to the motion. Therefore, the work done by a force can be described by the dot product of the force vector and the displacement vector. For example, several forces act on the skier in the diagram below.
Example: Xiao turns a crank to lower a bucket of water into a well. Determine the total work done on the bucket if the weight of the bucket is 15 N and the tension force in the rope is 13 N. The bucket rises a distance of 4.5 m while he is cranking.
The weight force does positive work on the bucket because the motion and the force are in the same direction.
The total work is the sum of the two individual amounts of work.
A total of 9.0 J of work is done on the bucket as the bucket moves downward into the well.
- The work done by a force on an object is given by the dot product of the force vector and the displacement vector representing the motion of the object. The diagram below shows Sanjay pulling a large crate across the floor. The four forces which act on the crate during this process are shown in the diagram below. Which of the four forces exert non-zero force on the crate?
- Le’andra is pulling her toy duck (mass 0.75 kg) at a constant speed of 3.0 m/s. The string she uses to pull the duck makes an angle of 42o above the horizontal and Le’andra keeps a constant tension in the string of 2.0 N. What is the amount of work done by the tension force when the duck is pulled forward a distance of 2.8 m?
- Beuford has once again taken Brynna to the park to play on the slide. If Brynna has a weight of 25 kg and if the slide has a 30o incline above the horizontal, what work is done by her weight as she slides down the 3.5 m incline? Remember that the weight force in newtons is equal to the product of the mass and the acceleration of gravity, 9.8 m/s2.
- The scientists at Fermi Lab in Chicago, IL use magnetic fields to direct beams of protons during their explorations of the submicroscopic structure of atoms and quarks. Both magnetic field strength and the velocity of the protons are vector quantities. Determine the force on a proton moving northward at 4.2 × 106 m/s through a magnetic field of 2.5 T oriented from east to west. (Note: protons are very tiny and are therefore able to move VERY fast.)
- Vector B⃗ represents the magnitude and direction of the magnetic field in a certain region. Vector v⃗ represents the velocity of a charged particle, q = 3.2 × 10-15 C, which moves through the magnetic field. The lengths of the vectors are such that the velocity vector is measured in m/s and the magnetic field vector is measured in tesla. What is the force experienced by the charged particle as it moves through the magnetic field?
- If two children, Rudolfo (weight = 210 N) and Jennifer (175 N), sit on either end of a see-saw as shown below. Rudolfo is 1.0 m from the pivot and Jennifer is 1.4 m from the pivot. What torques are exerted by the children on the see-saw?
- Exercise scientists and physical therapists use torque to analyze various exercises such as the triceps exercise shown in the diagram below. Here, the triceps muscle exerts a force on the elbow-end of the forearm, affecting the rotation of the forearm. If the triceps exerts a force of 17N, what torque is applied to the forearm?