• Multi Stage Rockets

    stage. Suppose that when the booster is discarded the speed of the rocket is u and the rocket is a distance R from the centre of the Earth. Take the mass of the unit as The force of gravity where is the mass of the Earth is the only force acting so...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3646-multi-stage-rockets.html

  • Particle Moving in a Horizontal Circle Attached to Two Rods

    in a horizontal circle, will produce forces in the rods which balance vertically and produce a force directed towards the centre of the circle which keep the partilce moving in the circle. We can find the forces in the rods by resolving vertically and...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3647-particle-moving-in-a-horizontal-circle-attached-to-two-rods.html

  • Particle on String Executing Circular Horizontal Motion

    we have to use the fact that the acceleration for a particle moving in a circle is of magnitude or and directed towards the centre of the circle. We apply (2) (2) divided by (1) gives Then Unlike as in simple harmonic motion the period does depend on...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3649-particle-on-string-executing-circular-horizontal-motion.html

  • Sphere Attached to a String Resting Against a Smooth Wall

    gives (2) Taking moments about B gives (3) Equating (1) and (3) gives so that the line of action of passes through the centre of the sphere.

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3654-sphere-attached-to-a-string-resting-against-a-smooth-wall.html

  • How Fidget Spinners Work

    fidget fidget spinner is a toy that you can set spinning b holding the centre between two fingers and giving the rim a push. When the fidget spinner starts to spin, it has angular momentum, with magnitude given by \[J= \omega I\] , directed...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/4698-how-fidget-spinners-work.html

  • Maximum Volume of Cylinder Inscribed In Sphere

    inscribed in a sphere of radius \[R\] ? Let the radius \[r\] of the cylinder base subtend an angle of \[\theta\] at the centre of the sphere then \[r=Rsin \theta\] and if the height of the cylinder is \[h\] then from the diagram \[h/2=R cos \theta...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5328-maximum-volume-of-cylinder-inscribed-in-sphere.html

  • A Circle in Polar Coordinates

    curve given in polar coordinates by \[r=2 sin \theta\] is a circle radius 1 with centre \[(0,1)\] . To see this let \[x=rcos \theta = 2sin \theta cos \theta , \; y=r sin \theta = 2 sin^2 \theta \] . Then \[\begin{equation} \begin{aligned} x^2+(y-1)^2 &=...

    https://astarmathsandphysics.com/a-level-maths-notes/fp2/5331-a-circle-in-polar-coordinates.html

  • Level Surfaces, Normals and Tangent Planes

    + \frac{ \partial f}{\partial z}|_{(x_0,y_0,z_0)}(z-z_0)=0\] . Example: For the level surface \[x^2+y^2+z^2=14\] (sphere centre the origin, radius \[\sqrt{14}\] ). The partial derivatives are \[2x, \; 2y, \; 2z\] respectively and at \[(1,2,3)\] these...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5343-level-surfaces-normals-and-tangent-planes.html

  • Volume of a Sphere Segment

    a sphere is cut by a plane, what will be the volume of the smaller segment? Take a sphere radius \[r\] , centre the origin, and let the sphere be cut by a horizontal plane \[y=a\] . We can consider the sphere segment to be generated by rotating an arc...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5359-volume-of-a-sphere-segment.html

  • Velocity and Acceleration of Piston Operated By a Flywheel

    forwards as the wheel rotates. If the radius of the flywheel is \[R\] and the length of the rod is \[l\] . and we let the centre of the flywheel be the origin, then from the diagram below The base of the piston is at \[x=R cos \theta + \sqrt{l^2- R^2...

    https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/5386-velocity-and-acceleration-of-piston-operated-by-a-flywheel.html

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