Answer:
Explanation:
Given that,
Mass of ball =M
Radius of ball is =R
Coefficient of kinetic friction =u
Initial speed of ball =Vo
The weight of the body acting downward is
W=Mg.
The normal reaction is acting upward and it is given as
From Newton law
N=W=mg
Frictional force is given as
Fr=µN
Fr= µmg
This is the only force on the x-axis
Then,
ΣF = ma
-Fr=ma
-µmg=ma
Divide through by m
a= -µg
The negative show that it is decelerating
So using equation of motion
V=u+at
Where V is final velocity,?
u is initial velocity =Vo
And a is acceleration =-µg
Then, the velocity at any point in time is given as
V = Vo - µgt
The angular acceleration that sets the ball rotating with increasing angular velocity in anticlockwise direction whose magnitude ω, at any instant t, is given by
ω = αt.
Also, V=ωR
V=αtR
To get angular acceleration
Further, the only force that produces a torque about the centre is fk. This torque is of magnitude fkR, acting in anticlockwise direction producing an anticlockwise angular acceleration, α, of the ball about its center given as
fk•R = Icm •α,
Icm for a sphere is 2/5MR²
µMg•R = (2/5)MR² α
Divide both side by MR
µg= (2/5)Rα
α = 5µg/2R.
From above
V = Vo - µgt, then, V=αtR
αtR= Vo - µgt
αtR + µgt= Vo
t(αR + µg)=Vo
t=Vo/(αR + µg)
Since α = 5µg/2R
t=Vo/( 5µg/2R • R + µg)
t = Vo/( 5µg/2+ µg)
t= Vo/(7µg/2)
t=2Vo/7µg
So, the time is given as
t = 2Vo/7µg
And the velocity at any time is given as
V = Vo - µgt,
