1. draw the trajectory of a point on a beam rotating. choose a coordinate system. choose a point on that trajectory that is not on a coordinate axis. draw vectors representing the position, velocity, and acceleration of that point.2. write equations for each component of the position vector, as a function of the distance of the point from the axis of rotation and the angle the vector makes with an axis of your coordinate system. next, calculate how that angle depends on time and the constant angular speed of the beam. sketch three graphs, (one for each of these equations) as a function of time. explain why one of the graphs increases monotonically with time, but the other two oscillate.3. using your equations for components of the position of the point, calculate an equation for each component of the velocity of the point. graph these two equations as a function of time. compare these graphs to those for the components of the position of the object (when one component of the position is at a maximum, for example, is the same component of the velocity at a maximum value? ) draw these components at the point you have chosen in your drawing; verify that their vector sum gives the correct direction for the velocity of the point.4. use your equations for the point’s velocity components to calculate its speed. does the speed change with time? should it? 5. use the equations for the point’s velocity components to calculate an equation for each component of the point’s acceleration. graph these two equations as functions of time, and compare to the velocity and position graphs. verify that the vector sum of the components gives the correct direction for the acceleration of the point you have chosen in your drawing. use the acceleration components to calculate the magnitude of the acceleration.6. for comparison, write down the expression for the acceleration of the point as a function of its speed and its distance from the axis of rotation. predictionwhat are you trying to calculate? restate the problem to clearly identify your objective.

  sequence  is a continuous and distinctive band of  stars  that appears on plots of stellar  color  versus  brightness. these color-magnitude plots are known as  hertzsprung–russell diagramsafter their co-developers,  ejnar hertzsprungand  henry norris russell. stars on this band are known as  main-sequence stars  or  dwarf stars. these are the most numerous true stars in the universe, and include the earth's  sun.after condensation and ignition of a star, it generates  thermal energy  in its dense core region through  nuclear fusion  of  hydrogen  into  helium. during this stage of the star's lifetime, it is located on the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and age. the cores of main-sequence stars are in  hydrostatic equilibrium, where outward thermal pressure from the hot core is balanced by the inward pressure of  gravitational collapse  from the overlying layers. the strong dependence of the rate of energy generation on temperature and pressure to sustain this balance. energy generated at the core makes its way to the surface and is radiated away at the  photosphere. the energy is carried by either  radiation  or  convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both.the main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. stars below about 1.5 times the  mass of the sun  (1.5  m☉) primarily fuse hydrogen atoms together in a series of stages to form helium, a sequence called the  proton–proton chain. above this mass, in the upper main sequence, the nuclear fusion process mainly uses atoms of  carbon,  nitrogen  and  oxygen  as intermediaries in the  cno cycle  that produces helium from hydrogen atoms. main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. below this mass, stars have cores that are entirely radiative with convective zones near the surface. with decreasing stellar mass, the proportion of the star forming a convective envelope steadily increases. main-sequence stars below 0.4  m☉undergo convection throughout their mass. when core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen.in general, the more massive a star is, the shorter its lifespan on the main sequence. after the hydrogen fuel at the core has been consumed, the star  evolves  away from the main sequence on the hr diagram, into a  supergiant,  red giant, or directly to a  white dwarf.

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