A large sphere rolls without slipping across a horizontal surface. The sphere ha

Question

A large sphere rolls without slipping across a horizontal surface. The sphere has a constant translational speed of 10 meters per second. a mass m of 25 kilograms, and a radius r of 0.2 meter. The moment of inertia of the sphere about its center of mass is I = 2mr^2/5. The sphere approaches a 25 degree incline of height 3 meters as shown above and rolls up the incline without slipping. Calculate the total kinetic energy of the sphere as it rolls along the horizontal surface. Calculate the magnitude of the sphere's velocity just as it leaves the top of the incline. Specify the direction of the sphere's velocity just as it leaves the top of the incline. Neglecting air resistance, calculate the horizontal distance from the point where the sphere leaves the incline to the point where the sphere strikes the level surface. Suppose, instead, that the sphere were to roll toward the incline as stated above, but the incline were frictionless. State whether the speed of the sphere just as it leaves the top of the incline would be less than, equal to, or greater than the speed calculated in (b). Explain briefly

Explanation / Answer

a)The total kinetic energy will be the sum of translational and rotational kinetic energies of the sphere.

KE = 1/2 m v^2 + 1/2 I w^2

1/2 I w^2 = 1/2 x 2/5 m r^2 x v^2/r^2 = 1/5 m v^2

KE = 0.5 x 25 x 10^2 + 0.2 x 25 x 10^2 = 1750 J

Hence, KE(total) = 1750 J

b)(i)from the conservation of energy

KE(total) = KE(leave) + PE(top)

KE(leaves) = Ke(total) - PE(top)

0.7 m v'^2 = 1750 - 25 x 9.8 x 3 = 1015 (KEtot = (0.5 + 0.2 m v^2))

v' = sqrt (1015/0.7 X 25) = 7.62 M/S

Hence, v' = 7.62 m/s

ii)theta = 25 deg above horizontal

c)V'x = 7.62 x cos25 = 6.9 m/s

v'y = 7.62 x sin25 = 3.22 m/s

from eqn of motion

h = ut + 1/2 at^2

-3 = 3.22 t - 4.9 t^2 (a = -g; h = -3 and u = v'y)

t = 1.18 s

D = v'x t = 1.18 x 6.9 = 8.14 m

Hence, D = 8.14 m

d)The speed of the ball will be greater in that case, as the ball would be sliding only instead of rolling and translational KE would be more.

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