All Work and No Play Learning Goal: To be able to calculate work done by a constant force directed at different angles relative to displacement If an object undergoes displacement while being acted upon by a force (or several forces), it is said that work is being done on the object. If the object is moving in a straight line and the displacement and the force are known, the work done by the force can be calculated as , where is the work done by force on the object that undergoes displacement directed at angle relative to . Note that depending on the value of , the work done can be positive, negative, or zero. In this problem, you will practice calculating work done on an object moving in a straight line. The first series of questions is related to the accompanying figure. In the next series of questions, you will use the formula to calculate the work done by various forces on an object that moves 160 meters to the right. Understanding Work and Kinetic Energy Learning Goal: To learn about the Work-Energy Theorem and its basic applications. In this problem, you will learn about the relationship between the work done on an object and the kinetic energy of that object. The kinetic energy of an object of mass moving at a speed is defined as . It seems reasonable to say that the speed of an object--and, therefore, its kinetic energy--can be changed by performing work on the object. In this problem, we will explore the mathematical relationship between the work done on an object and the change in the kinetic energy of that object. First, let us consider a sled of mass being pulled by a constant, horizontal force of magnitude along a rough, horizontal surface. The sled is speeding up. Let us now consider the situation quantitatively. Let the mass of the sled be and the magnitude of the net force acting on the sled be . The sled starts from rest. Consider an interval of time during which the sled covers a distance and the speed of the sled increases from to . We will use this information to find the relationship between the work done by the net force (otherwise known as the net work) and the change in the kinetic energy of the sled. Here is a simple application of the Work-Energy Theorem. Work-Energy Theorem Reviewed Learning Goal: Review the work-energy theorem and apply it to a simple problem. If you push a particle of mass in the direction in which it is already moving, you expect the particle's speed to increase. If you push with a constant force , then the particle will accelerate with acceleration (from Newton's 2nd law). Pulling a Block on an Incline with Friction A block of weight sits on an inclined plane as shown. A force of magnitude is applied to pull the block up the incline at constant speed. The coefficient of kinetic friction between the plane and the block is . Now the applied force is changed so that instead of pulling the block up the incline, the force pulls the block down the incline at a constant speed. Dragging a Board A uniform board of length and mass lies near a boundary that separates two regions. In region 1, the coefficient of kinetic friction between the board and the surface is , and in region 2, the coefficient is . The positive direction is shown in the figure. Baby Bounce with a Hooke One of the pioneers of modern science, Sir Robert Hooke (1635-1703), studied the elastic properties of springs and formulated the law that bears his name. Hooke found the relationship among the force a spring exerts, , the distance from equilibrium the end of the spring is displaced, , and a number called the spring constant (or, sometimes, the force constant of the spring). According to Hooke, the force of the spring is directly proportional to its displacement from equilibrium, or . In its scalar form, this equation is simply . The negative sign indicates that the force that the spring exerts and its displacement have opposite directions. The value of depends on the geometry and the material of the spring; it can be easily determined experimentally using this scalar equation. Toy makers have always been interested in springs for the entertainment value of the motion they produce. One well-known application is a baby bouncer,which consists of a harness seat for a toddler, attached to a spring. The entire contraption hooks onto the top of a doorway. The idea is for the baby to hang in the seat with his or her feet just touching the ground so that a good push up will get the baby bouncing, providing potentially hours of entertainment. Delivering Rescue Supplies You are a member of an alpine rescue team and must project a box of supplies, with mass , up an incline of constant slope angle so that it reaches a stranded skier who is a vertical distance above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient . Work Done by a Spring Consider a spring, with spring constant , one end of which is attached to a wall. The spring is initially unstretched, with the unconstrained end of the spring at position . Exercise 6.41 A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 33.0 above the horizontal. The glider has mass 7.00×10−2 . The spring has 680 and negligible mass. When the spring is released, the glider travels a maximum distance of 1.60 along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. Problem 6.69 A small block with a mass of 0.180 is attached to a cord passing through a hole in a frictionless, horizontal surface (the figure ). The block is originally revolving at a distance of 0.50 from the hole with a speed of 0.20 . The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.10 . At this new distance, the speed of the block is observed to be 1.00 . Problem 6.73 You and your bicycle have combined mass 80.0 . When you reach the base of a bridge, you are traveling along the road at 7.50 (the figure ). At the top of the bridge, you have climbed a vertical distance of 5.20 and have slowed to 3.00 . You can ignore work done by friction and any inefficiency in the bike or your legs. Problem 6.81 A 5.00- block is moving at 6.00 along a frictionless, horizontal surface toward a spring with force constant =500 that is attached to a wall (the figure ). The spring has negligible mass. A Car with Constant Power The engine in an imaginary sports car can provide constant power to the wheels over a range of speeds from 0 to 70 miles per hour (mph). At full power, the car can accelerate from zero to 31.0 in time 1.30 .