Question

"Two uniform identical solid spherical balls each of mass M and radius R" and moment of inertia about its center 2/5 MR2 are released from rest from the same height H above the horizontal ground. Ball A falls straight down (no air resistance) while B rolls down the inclined plane without slipping. Which ball has the greater TME at the bottom of the incline?

Answer 1

Answer:

he sphere that uses less time is sphere A

Explanation:

Let's start with ball A, for this let's use the kinematics relations

        v² = v₀² - 2g (y-y₀)

indicate that the sphere is released therefore its initial velocity is zero and when it reaches the floor its height is zero y = 0

         v² = 0 - 2 g (0- y₀)

         v = $\sqrt{2g y_o}$

         v = $\sqrt{2 \ 9.8\ H}$

         v = 4.427 √H

Now let's work the sphere B, in this case it rolls down a ramp, let's use the conservation of energy

starting point. At the highest point, before you start to move

         Em₀ = U = m g y

final point. At the bottom of the ramp

         Em_f = K = ½ m v² + ½ I w²

notice that we include the kinetic energy of translation and rotation

energy is conserved

          Em₀ = Em_f

          mg H = ½ m v² + ½ I w²

angular and linear velocity are related

          v = w r

          w = v / r

the momentorot of inertia indicates that it is worth

          I = $\frac{2}{5}$ m r²

we substitute

           m g H = ½ m v² + ½ ($\frac{2}{5}$  m r²) ($\frac{v}{r}$ )²

           gH = $\frac{1}{2}$  v² + $\frac{1}{5}$  v² = $\frac{7}{10}$  v²

           v = $\sqrt{\frac{10}{7} \ g H}$

           v = $\sqrt{ \frac{10}{7} \ 9.8 \ H}$

           v=3.742 √H

Taking the final speeds of the sphere, let's analyze the distance traveled, sphere A falls into the air, so the distance traveled is H.  The ball B rolls in a plane, so the distance (L) traveled can be found with trigonometry

           sin θ = H / L

           L = H /sin θ

we can see that L> H

In summary, ball A arrives with more speed and travels a shorter distance, therefore it must use a shorter time

Consequently the sphere that uses less time is sphere A