NCERT

Those who are preparing for any competive exam such as JEE Main, NDA and others really need to memorise formulas. Check the important formulas below;  

Circle

• Standard equation (center at origin): $x^2+y^2=r^2$
  $x^2 + y^2 = r^2$

• General form: $x^2 + y^2 + 2gx + 2fy + c = 0$

• Center = (–g, –f), Radius = ?(g² + f² – c)

 Parabola: 

• Standard form (horizontal axis): $y^2=4ax$
  $y^2 = 4ax$

• Standard form (vertical axis): $x^2=4ay$
  $x^2 = 4ay$

• Vertex: (0, 0)

• Focus: (a, 0) or (0, a)

• Directrix: x = –a or y = –a

• Latus rectum = 4a

Ellipse: Check all the important topics related to this topic below;

• Standard form (major axis along x-axis): 

  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where a > b

  • Eccentricity: $e=\sqrt{1?\frac{b^2}{a^2}}$
    $e = \sqrt{1 - \frac{b^2}{a^2}}$

• Foci: (±ae, 0)

• Latus rectum: $\frac{2b^2}{a}$

$$Hyperbola: Students can check important formulae related to this topic below;

• Standard form (transverse axis along x-axis):
  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Eccentricity:
  $e = \sqrt{1 + \frac{b^2}{a^2}}$

• Foci: (±ae, 0)

• Latus rectum: $\frac{2b^2}{a}$

• Asymptotes:
  $y=\pm \frac{b}{a}x$

According to the mechanical properties of fluids class 11, the liquid's ability to flow in narrow spaces in opposition to external forces like gravity, without assistance is called the capillarity or capillary action.
When the liquid rises or falls in a narrow tube (capillary tube), it is called capillarity. It results from the adhesive forces between the walls of the tube and the liquid and the cohesive forces within the liquid. The liquid rises in a tube and forms a concave meniscus when the cohesive forces are weaker than the adhesive forces. The liquid will be depressed in the tube, if the cohesive forces are stronger than the adhesive forces, forming a convex meniscus.
Several factors determine the height of the liquid column in the capillary tube including the contact angle between the tube and liquid, the surface tension of the liquid and the density of the liquid, the radius of the capillary tube, and the acceleration due to gravity. A higher surface tension and a narrower tube lead to greater rise or fall.

The property of a liquid surface which lets it behave like a stretched elastic membrane is called surface tension. It is the force per unit length acting in the plane of the interface between any medium such as air and liquid, tending to contract the surface area. In the cohesive forces between the liquid molecules lies the molecular origin of surface tension. The bulk of liquid contains the molecules which are surrounded by other liquid molecules. It experiences attractive forces in all directions which balance out. The molecules at the surface experience net inward force due to no liquid molecules present above them to balance the cohesive forces from within. The inward forces allow the surface to contract to the least possible area. The example of surface tension is the small insects walking on the water, the rise of water in a capillary tube, and the formation of spherical droplets of liquids.

Viscosity is the measure of the resistance of fluid towards the shear stress when it is applied for fluid to flow or deformation. Viscosity describes the internal friction within a fluid. A fluid with high viscosity has more internal friction and hence it resists motion. If one pours water and honey, honey flows slower because it has higher viscosity. This affects the ease of a fluid with which it moves, higher viscosity means slow flow due to greater resistance.

A fluid's viscosity can be influenced by various factors. It mostly decreases with increasing temperature as the intermolecular forces are weakened by the increased kinetic energy. In the case of the gases, the increasing temperature generally decreases the viscosity because there are more frequent molecular collisions leading to greater internal friction and greater momentum transfer. The presence of dissolved substances, impurities, and types of fluid also affects the viscosity.

According to the NCERT Solutions for Class 11 Physics Chapter 9 Mechanical Properties of Fluids, when an object is immersed in the fluid, the fluid opposes its weight and exerts an upward force, called buoyancy. It is due to the difference in the pressure of the fluid at different depths, at the bottom of a submerged object, the fluid pressure is more than the pressure at its top. Archimedes' principle is used to quantify this buoyant force. It states that when an object is fully or partially immersed in a fluid, the upward buoyant force it experiences is equal to the fluid weight displaced by the body. To understand whether the object will sink or float, we should compare the buoyant force with the object's weight. The object will float if the buoyant force is equal to or greater than the weight of the object. It can float partially or fully depending on the buoyant force and the object's weight. When the object's weight is more than the buoyant force, the object will sink into the fluid.

According to Pascal's Law when in a confined fluid, pressure is applied, the pressure increase is transmitted undiminished and equally in all directions throughout the fluid. In hydraulic systems, this principle is crucial where a small force is applied at one point resulting in more force at another point. Examples include hydraulic jacks, braking systems in vehicles, and hydraulic lifts. Pascal's Law works because fluids are virtually incompressible, and when pressure is applied to the fluids, it is distributed uniformly. The law shows that to multiply force, how fluids can be used, which is the fundamental principle behind many mechanical devices.