Arod of length l and mass m has a nonuniform mass distribution. the linear mass density (mass per length) is λ=cx2, where x is measured from the center of the rod and c is a constant.a. what are the units of c? mastering physics says the answer is: kg/m^3 but i don't understand where they got this answer.b. find the expression for c.mastering physics says the answer is 12m/l^3 --> no clue where they got this answer! c. find the expression for the moment of inertia of the rod for rotation about an axis through the center.

Explanation:

λ=c x²

c = λ / x²

λ is mass / length

so its dimensional formula is ML⁻¹

x is length so its dimensional formula is L

c = λ / x²

= ML⁻¹ / L²

= ML⁻³

B )

We shall find out the mass of the rod with the help of given expression of mass per unit length and equate it with given mass that is M

The mass in the rod is symmetrically distributed on both side of middle point.

we consider a small strip of rod of length dx at x distance away from middle point

its mass dm = λdx = cx² dx

By integrating it from -L to +L we can calculate mass of whole rod , that is

M = ∫cx² dx

= [c x³ / 3] from -L/2 to +L/2

= c/3 [ L³/8 + L³/8]

M = c L³/12

c = 12 M L⁻³

C ) Moment of inertia of rod

∫dmx²

= ∫λdxx²

= ∫cx²dxx²

= ∫cx⁴dx

= c x⁵ / 5 from - L/2 to L/2

= c / 5 ( L⁵/ 32 +L⁵/ 32)

= (2c / 160)L⁵

= (c / 80) L⁵

= (12 M L⁻³/80)L⁵

= 3/20 ML²

=

=

m = 63.7 kg

Explanation:

As we know that when mass connected to the bungee cord stretch the string then the gravitational potential energy of the person will convert into potential energy of the string at the end

now here we know that when person jump from the top and reach at the end then loss in gravitational potential energy is given as

now when it is at the end of the motion stretch in the string will be

now potential energy of string is given as

now by energy conservation we have