Tutorial No. 1 PDF

# Tutorial No-1 ## 9.1 Define Uniform Circular Motion During circular motion, if the speed of the particle remains constant, it is called uniform circular motion. * The cause responsible for this motion is called centripetal acceleration. $a = -w^2r$ * For a velocity with constant magnitude: $...

# Tutorial No-1 ## 9.1 Define Uniform Circular Motion During circular motion, if the speed of the particle remains constant, it is called uniform circular motion. * The cause responsible for this motion is called centripetal acceleration. $a = -w^2r$ * For a velocity with constant magnitude: $a = w^2r = \frac{v^2}{r} = wv$ ## 9.2 Non-Uniform Circular Motion For non-uniform circular motion, the magnitude of centripetal acceleration is not constant. * This is because the magnitude of tangential velocity is changing during non-uniform circular motion, and the corresponding angular velocity is also changing at every instant. * In this case, the particle is still in the same plane, and the direction of acceleration is always along the axis of rotation. * When the particle’s speed ω is increasing $\overrightarrow{a}$ and $\overrightarrow{w}$ are both in the same direction. * When the particle's speed ω is decreasing $\overrightarrow{a}$ and $\overrightarrow{w}$ are in opposite directions. ## 9.3 Derive an expression that relates angular momentum with the angular velocity of a rigid body. * **The diagram above shows a rigid object with many particles, rotating with a constant angle speed w about its axis perpendicular to the plane of paper. For the sake of simplicity of the action, we consider the object to be consisting of N number of particles. All the particles perform the axis of the rotation with the same angular velocity w, but with different linear speeds: $V_1 = r_1w$, $V_2 = r_2w$, $V_3 = r_3w$, ... $V_N=r_Nw$ * **The direction of each individual velocity.** * **Destiny which belongs to centripetal force (cp) and centrifugal force (cf) ** ### Centripetal Force (cp) * **The cause responsible for circular motion is the centripetal acceleration (cp) - it’s a radial acceleration. * The direction of the centripetal force is towards the center of the circle. * Centripetal force = $mw^2r$ ### Centrifugal Force (cf) * Centrifugal force is an unaccelerated force, but not an imaginary force. * The direction of the centrifugal force is away from the center. ### Centripetal Force and Centrifugal Force * When determining the motion round in a circular motion, we feel outward pull. * This is because the particle is under the influence of a real force, because of which we experience an outward pull. * The real force acting upon the body is two-fold and tends towards the center. * **Centripetal force:** This is the real force that keeps the body in a circular path. It is the force that is responsible for the body’s centripetal acceleration. * **Centrifugal Force: ** This is the imaginary force that we experience. It is the force that is the result of the body’s inertia. We feel like we are being pushed outward, but in reality, this is just the body’s tendency to maintain its straight linear motion. * The difference between the centripetal and centrifugal forces is that the centripetal force is always present, while the centrifugal force is only present if the body is accelerating. * Centripetal force is a real force, and we can measure it, but the centrifugal is not a real force. ## 9.4 Describe the terms parallel axes * In order to apply the theorem of parallel axes to any object, we need two parallel axes to each other, with one of them passing through the entire mass of the object. The image shows a rigid body and a line of a rigid body with mass M. The coordinate axis is drawn with origin O. The axis A’ B’ passes through the point O with the coordinate axis. The axis AC’ passes through the center of mass C of the object, and it is parallel to the axis A’B’ with a distance h from it. * Let, $I_c$ = Moment of inertia of the object about axis AC’ * Let, $I_o$= Moment of inertia of the object about axis A’B’ * $D_o$ = distance of an individual element * $D _c$ = distance of an individual element * So we have: $ I_O = \int{(D_O)^2dm}$ $ I_o = \int {[(ON)^2+(NA)^2 ]dm}$ $ I_o = \int {[(DN)^2+(NC)^2 +2NC.h. (cot(θ))^2]dm}$ $ I_o = \int {[(D_c )^2 + 2NC.h.h^2]dm}$ $ I_o = \int {(D_c)^2dm} + 2\int{NC.h^2}.dm$ $ I_o = I_C + 2MCh^2$ $ I_o = I_C + Mh^2 $ where: * $I_C = \int (D_c)^2 dm $ * $Mh^2 = \int 2NC.h^2.dm$ = mass * h2 and h is the distance of a point from the center of mass. This is the mathematical proof of the theorem of parallel axes. ## Tutorial No: 2 ## Mechanical Properties of Fluids. ## 9.1 State the properties of an ideal fluid: * **Incompressible:** The density is constant. * *Note: In the real world, all fluids are compressible to some extent, but some, such as water, are much less compressible than others, such as air.* * **Irrotational:** The fluid does not rotate. * **Smooth:** The flow is smooth. * **No turbulence:** There is no internal friction in the flow. * **Non-viscous:** There is no internal friction in the flow. * **Steady:** The velocity at each point is constant in time. ## 9.2 Define pressure and give its SI unit and dimension: * The normal force exerted by a fluid per unit area of contact is called the pressure of the fluid: $P = F/A$ * **SI Units:** N/m² * Pascal (Pa) = 1N/m² * **Dimension:** [ML⁻¹T⁻²] ## 9.3 State any two applications of Pascal's Law: * **Hydraulic lift:** Pascal's Law is the basis for hydraulic lift systems. * **Manometer:** Pascal's Law is the basis for manometer designs. ## 9.4 Molecular Theory of Surface Tension: * Two molecules attract each other. This force between them is called the intermolecular force. * **Types of intermolecular forces:** * **Cohesive force:** This force is between the same type of molecule. * **Adhesive force:** This force is between the different types of molecules. * This cohesive force between the same kind of molecule is what causes surface tension. ## 9.5 Derive an expression for capillary rise for a liquid having a concave meniscus (water) The pressure due to the liquid (water) column of height h must be equal to the pressure difference 2T/R due to concavity. * *hpg* = *2T/R* * ... *h* = *2T*cos(θ) / (*pg*r). Let: * *R* be the radius of the meniscus. * *r* be the radius of the capillary tube. * θ be the angle of contact of the liquid. **The radius of contact curvature R of the meniscus is given by:** * *R* = *r*/cos(θ) * ... *h* = *2T*cos(θ) / (*pg*r*) = *2T*cos(θ)² / (*pg*R*) ## 9.6 Distinguish between Streamline flow and Turbulent flow: ### Streamline flow: * Smooth flow with velocity small, less than a critical velocity (lower limit). * The flow is called streamline or laminar flow of a fluid. ### Turbulent flow: * The flow is irregular and instant. * The velocity is greater than the critical velocity, resulting in turbulent flow. ## 9.7 Points to compare between streamline and turbulent flow: ### Streamline flow: * The velocity of the fluid at any point is constant. * Two streamlines can never intersect. * Streamline flow over a plane surface can be assumed to be divided into a number of plane uniform cross sections. * All streamlines will be parallel to the axis of the tube. ### Turbulent flow: * The velocity of the fluid at any point is not constant. * In turbulent flow, the fluid may have rotational motion which gives rise to eddies in the flow. * The flow is random and unpredictable. ## 9.8 State any two application of Bernoulli’s theorem: * **Lifting up of an airplane:** The streamlines of air are crowded above the wings. The air above the wings moves faster than below it, causing the wings to be lifted up. * **Venturi tube:** A venturi tube is used to measure the speed of flow of a fluid through the tube. The tube is constructed with a constriction. The fluid flows ideally and the pressure difference between the two verticle tubes connected to the venture is, and the difference in the height of the fluid column is h: * The pressure difference between the two vertical tubes connected to the venturi tube is ( P¹ - P² = pgh). * Using the above equation, one can get a direct relationship between the change in the velocity of the fluid and pressure. ## 9.9 Define cohesive force and adhesive force: * **Cohesive force:** This is the force of attraction between the molecules of the same substance due to their mutual attraction. * **Adhesive force:** This is the force of attraction between the molecules of different substances.