HCI 04 Forces Lecture Notes

Hwa Chong Institution (College) H2 Physics C1 2023 Chapter 4 Forces “The same principle that allows an airplane to rise off the ground by creating lift under its wings is used in reverse in F1 cars to generate an additional ‘downward force’ to press the race car against the surface of the track. This increases the contact force between the tires and the road surface, allowing the car to turn corners at amazing speeds. F1 cars achieve down ward force-to-weight ratio of 1:1 at about 125 km/h. At 190 km/h (118 mph) the ratio is roughly 2:1.” - Anonymous Formula One Expert

Hwa Chong Institution (College) H2 Physics C1 2023 2 TOPIC 4: Forces H2 Physics Syllabus 9749 Forces Learning Outcomes Students should be able to: Types of force (a) recall and apply Hooke’s law (F = kx, where k is the force constant) to new situations or to solve related problems (b) describe the forces on a mass, charge and current-carrying conductor in gravitational, electric and magnetic fields, as appropriate (covered later in relevant chapters) (c) show a qualitative understanding of normal contact forces, frictional forces and viscous forces including air resistance (no treatment of the coefficients of friction and viscosity is required) (partly covered in Kinematics) Centre of gravity (d) show an understanding that the weight of a body may be taken as acting at a single point known as its centre of gravity (covered in Dynamics) Turning effects of forces (e) define and apply the moment of a force and the torque of a couple (f) show an understanding that a couple is a pair of forces which tends to produce rotation only (g) apply the principle of moments to new situations or to solve related problems Equilibrium of forces (h) show an understanding that, when there is no resultant force and no resultant torque, a system is in equilibrium (i) use a vector triangle to represent forces in equilibrium Upthrust (j) derive, from the definitions of pressure and density, the equation p = ρgh (k) solve problems using the equation p = ρgh (l) show an understanding of the origin of the force of upthrust acting on a body in a fluid (m) state that upthrust is equal in magnitude and opposite in direction to the weight of the fluid displaced by a submerged or floating object (n) calculate the upthrust in terms of the weight of the displaced fluid (o) recall and apply the principle that, for an object floating in equilibrium, the upthrust is equal in magnitude and opposite in direction to the weight of the object to new situations or to solve related problems.

Hwa Chong Institution (College) H2 Physics C1 2023 3 Contents Topic 4: Forces 4.1 Hooke’s Law .............................................................................................................. 4 4.2 Upthrust or Buoyant Force ........................................................................................ 4 4.2.1 Pressure due to Fluid ........................................................................................ 4 4.2.2 Derivation of the equation p = ρgh ..................................................................... 5 4.2.3 Upthrust ............................................................................................................. 6 4.3 Translational Equilibrium ........................................................................................... 8 4.4 Moment of a Force .................................................................................................... 9 4.5 Rotational Equilibrium and Principle of Moments .................................................... 10 4.6 Static Equilibrium for Rigid Extended Bodies........................................................... 11 4.7 Three-Force Systems .............................................................................................. 12 Appendix I Contact force and frictional force ............................................................... 15 Appendix II viscous force .............................................................................................. 16 Appendix III Buoyancy of a submarine .......................................................................... 17 Tutorial 4 Forces .............................................................................................................. 18 Self-Review Questions ................................................................................................ 18 Discussion Questions .................................................................................................. 20 Videos of Lecture Examples can be found at https://youtube.com/playlist?list=PL_b5cjrUKDlaffEZq6U4qTf1LEZbyIgF-

Hwa Chong Institution (College) H2 Physics C1 2023 4 Let us begin by introducing some forces that we will encounter in this chapter. 4.1 Hooke’s Law When we try to extend a spring by pulling it apart or compressing it with both hands, each of our hands will be subjected to an opposing force by the spring. We refer to this force as the tension or compression in the spring. For our purposes, we usually consider light springs of negligible mass. In 1676, Robert Hooke stated an empirical law that allows us to calculate the magnitude of this force. # Note that, in general, all elastic materials obey Hooke’s Law within their elastic limits. The law is not restricted to springs. 4.2 Upthrust or Buoyant Force 4.2.1 Pressure due to Fluid If an object is immersed in a fluid, the fluid will press on the surface of the object. The normal force per unit area of the surface is referred to as the pressure due to the fluid. For a cylinder that is filled with water, the water pressure on the base of the cylinder will increase if more water is added and the water level rises. Similarly, if a small sheet of metal is dropped into the cylinder, the pressure on the surface of the metal sheet will increase as it sinks deeper. Apart from water, we also experience atmospheric pressure due to the air above us. The atmospheric pressure we experience as we climb up a mountain will decrease as we ascend higher. Hooke's Law states that the magnitude of force F exerted by a spring# on a body attached to the spring is proportional to the extension x of the spring from its natural length provided the proportional limit of the spring is not exceeded. kxF = k = constant of proportionality (also referred to as force constant or spring constant or stiffness of spring) Example 1 (N94/I/23) A spring, obeying Hooke’s Law, has an unstretched length of 50 mm and a spring constant of 400 N m-1. What is the tension in the spring when its overall length is 70 mm? [Answer: 8.0 N]

Hwa Chong Institution (College) H2 Physics C1 2023 5 4.2.2 Derivation of the equation p = ρgh Consider a column of fluid as shown in the diagram. This column of fluid is pressed inwards on all sides by the rest of the fluid in the beaker. Let  (Greek letter rho) = density of the fluid, A = surface area of the bottom of the column, h = depth from the surface of the fluid, V = volume of the fluid column, m = mass of the fluid column. The weight of the fluid is pressing down on the bottom surface A. Since pressure is force over area, the pressure due to the weight of the fluid column is 𝑝 = 𝑚𝑔 𝐴 . Recall that density is mass divided by volume, ρ = m / V. Rearranging, we find that m = ρ V, from which we get 𝑝 = 𝜌𝑉𝑔 𝐴 . Note further that the volume of the fluid column is equal to the height of the column multiplied by the surface area, V = h A. Thus, we find that V / A = h, from which 𝑝 = 𝜌𝑔ℎ. Hence, the average pressure due to the weight of a fluid column is p = ρgh. Note that we also have atmos