Forces on Systems Many real-world systems involve multiple objects and forces acting in different directions. To analyze motion in these complex systems, the first step is to define what objects to include in the system. Identifying internal and external forces that act on objects in the system is next. The external forces acting determine the motion of each object. Once the system is defined, a free-body diagram and force-acceleration equation may be constructed for each object, and these will help to determine a model for the whole system. It is also important to determine whether or not the system is in a state of equilibrium. When in dynamic equilibrium, objects move with constant velocity by inertia alone. When in static equilibrium, objects do not move. Whether in dynamic equilibrium or static equilibrium, the net force acting on the object is zero and torque is zero. Torque is the measure of how effective a force is at producing a rotation, and it is the product of the magnitude of the force and the lever arm (the distance from the force to the point of rotation). Torque 1 = torque r = lever arm F = force The acceleration, net force, and torque for a system in equilibrium can be set equal to zero. 3'} Static Equilibrium N - - This system is in static equilibrium. Forces sum to zero and torques sum \\ to zero. 1' f ' ' '|H,!"' (Luff, M, 0,. 1 rs rL WSmall Note the sign convention ngg established for the clockwise + and Balance Forces Balance Torque; countercldclgwise (_) 2F = 0 1 = 0 directions. NWsWL=U +WerWLrL=0 Use Math How do the lever arm distances to each dog compare if the large dog is four (4) times more massive than the small dog? Show your math reasoning. if Mathematical Practices: Model with Mathematics Solving motion problems for complex systems involves identifying important quantities, mapping their relationships using diagrams, and writing force-acceleration equations. Important quantities include masses, forces, and torques. They might also include quantities that appear in the equations for various force types, such as the coefficient of friction for a material, the acceleration due to gravity, or a spring constant. Mapping relationships involves constructing a diagram of the entire system, then deconstructing the system by drawing a free-body diagram for each object. This process helps you determine if the system is in equilibrium or if the motion of objects is changing. Using Mathematical Models to Represent a Pulley System Observe how identifying important quantities and mapping their relationship to each other is needed to represent the pulley system. Important quantities in the pulley system include: 0 Mass of the furniture dolly (mD) 0 Mass of the bag (m3) 0 Acceleration due to gravity (g) 0 Tension (T) 0 Normal on the furniture dolly (N) The relationships among these quantities is understood by using them to label forces in free-body diagrams. The diagrams help you write specific force-acceleration equations for each object, one for each direction of motion (x and y). Such equations can help you answer questions about the pulley system. Forces on Systems 23 3 '1. Dimensional Review SEP Analyzing and Interpreting Data A meter stick is balanced at its center. The table shows the weight and distance from the center of two weights on the meter stick. Will it be in static equilibrium? Explain your reasoning using torque. Weight Weight (N) I Distance to Center (m) A 0.50 N 0.2 m to the right B 0.30 N 0.4 m to the left 2. 3. 24 DCI Forces and Motion A rope and pulley are used to lower a piano with a mass of 450.0 kg. The rope is tied to a weighted cart on a horizontal surface with a coefficient of friction of 0.16. The rope is then wrapped over the pulley out of a window and tied to the piano. a. If the weighted cart has a greater mass than the piano, will the system move? Explain your reasoning. b. If the piano has a greater mass, how will the tension and acceleration of the weighted cart change? c. If the piano is lowered at a constant velocity, what is the tension in the rope and the mass of the weighted cart? f CCC Cause and Effect A meter stick is balanced at its center using a support. Two weights of different masses are hung on either side. The meter stick stays balanced and does not move. Which of the following statements accurately describes what is happening? a. Weight A provides a greater torque. b. Weight B has a greater mass. c. Both weights are the same mass as the stick is in static equilibrium. d. Both weights provide the same torque as the stick is in static equilibrium. Investigation 2 Forces Skills Practice 4. A meter stick is balanced at its center using a support. A 0.45-N weight is hung on the meter stick a distance 0.30 m from the center to the right. Where should you place a 0.20-N weight such that the meter stick is in static equilibrium? f 5. An Atwood machine is a mechanical system consisting of two objects connected via a rope over a pulley. Suppose two objects, A and B, are connected by a rope. The rope is hung over a frictionless pulley with an object hanging on either side. The system is allowed to move freely. Object A has a mass of 21 .0 kg and object B has a mass of 15.0 kg. If object A is released from rest, determine the magnitude of the acceleration and tension for object B. Assume the rope and pulley are massless. y' 6. A meter stick is balanced at its center using a support. Two weights of unknown mass are hung on either side of the meter stick, as shown in the image. Which of the following statements accurately describes the relationship between the two weights? L r 2 lA_L\\ 3. WA = WB b. WA = 2W3 1:. WA = %WB d. WA = %WB Forces on Systems 25