PLEASE PROVIDE EXPLANATION FOR D AND E

THANK YOU!

d) 14.1 m

e) 8.57 m/s

Explanation:

(a) There are three forces on the ball:

Normal force Fn pushing up,

Weight force Fg pulling down,

Kinetic friction Fk pushing left (resisting motion).

FBD is the correct free body diagram.

(b) Sum of forces in the y direction:

∑F = ma

Fn − Fg = 0

Fn = Fg

Sum of forces in the x direction:

∑F = ma

-Fk = ma

-Fn μk = ma

-mg μk = ma

a = -g μk

a = -(10 m/s²) (0.25)

a = -2.5 m/s²

(c) Sum of torques about the ball's center:

∑τ = Iα

Fk r = ⅖ mr² α

mg μk r = ⅖ mr² α

g μk = ⅖ r α

α = 5g μk / (2r)

α = 5 (10 m/s²) (0.25) / (2 × 0.2 m)

α = 31.25 rad/s²

(d) The velocity and angular velocity at time t is:

v = at + v₀

v = -2.5t + 12.0

ω = αt + ω₀

ω = 31.25t + 0

When the ball begins to roll without slipping, v = ωr.

-2.5t + 12.0 = (31.25t) (0.2)

-2.5t + 12.0 = 6.25t

12.0 = 8.75t

t = 1.37

The distance the ball moves is:

Δx = v₀ t + ½ at²

Δx = (12) (1.37) + ½ (-2.5) (1.37)²

Δx = 14.1 m

(e) v = at + v₀

v = (-2.5 m/s²) (1.37 s) + 12.0 m/s

v = 8.57 m/s