A9.76-m ladder with a mass of 22.6 kg lies flat on the ground. a painter grabs the top end of the ladder and pulls straight upward with a force of 254 n. at the instant the top of the ladder leaves the ground, the ladder experiences an angular acceleration of 1.99 rad/s2 about an axis passing through the bottom end of the ladder. the ladder's center of gravity lies halfway between the top and bottom ends.
a) what is the net torque acting on the ladder?
b) what is the ladder's moment of inertia?

The definition of torque and Newton's second law for rotation allows to find the results for the questions about the rotational motion of the ladder are:

   A) The net torque is:  τ = 1397.1 N m

   B) the moment of inertia of the ladder with respect to the pivot point is:

        I = 702.1 Kg m²

Torque is the product of force for the distance perpendicular to the point of application and is the effect of force on angular motion.

        τ = F x r

 

Where bold indicates vectors, τ is torque, F is force, and r is distance to the pivot point.

The module of this expression can be written explicitly

           τ = F r sin θ

where the quantity (r sin θ) is called the perpendicular distance or arm.

Part A)

In this case they indicate that the applied force is F = 254 N at the ends of the ladder at L = 9.76 m from the turning point and the center of mass is at the middle point of the ladder.

In the attached we can see a diagram of the forces, where the reference system at the turning point and the counterclockwise as positive.

        ∑τ = F L - W L / 2

Weight is:   W = mg

         

        Σ τ = F L - mg L / 2

        Σ τ = 254 9.76 - 22.6 9.8 9.76 / 2

        Σ τ = 1397.1 Nm

Part B)

Newton's second law for rotational motion establishes the relationship between the net torque and the product of the moment of inertia and the angular acceleration.

       Σ τ = I α

They indicate the value of the angular acceleration α= 1.99 rad/a²

       I =  

Let's calculate

      I =  

      I = 702.1 kg m²

In conclusion using the definition of torque and Newton's second law for rotation we can find the results for the questions about the rotational motion of the ladder are:  

   A) The net torque is τ = 1397.1 Nm

   B) The moment of inertia of the ladder with respect to the pivot point is:

         I = 702.1 Kg m²

Learn more here: link

Part a)

Part b)

Explanation:

Part a)

Torque on the rod is due to two forces

(i) Due to weight of the rod at mid point

(ii) Due to external force applied at the end

Now we have

Part b)

As per Newton's law we know that

now we have

  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.