To solve this problem it is necessary to apply the concepts related to the kinetic energy expressed in terms of simple harmonic movement, as well as the concepts related to angular velocity and acceleration and linear acceleration and velocity.
By definition we know that the angular velocity of a body can be described as a function of mass and spring constant as

Where,
k = Spring constant
m = mass
From the given values the angular velocity would be


The kinetic energy on its part is expressed as

Where,
A = Amplitude
= Angular Velocity

PART A) Replacing previously given values the energy in the system would be



PART B) Through the amplitude and angular velocity it is possible to know the linear velocity by means of the relation



PART C) Finally, the relationship between linear acceleration and angular velocity is subject to



Answer:
The velocity is
Explanation:
From the question we are told that
The mass of the ball is 
The radius is 
The force is 
The speed of the ball is 
Generally the kinetic energy at the top of the circle is mathematically represented as

=>
=>
Generally the work done by the force applied on the ball from the top to the bottom is mathematically represented as

Here d is the length of a semi - circular arc which is mathematically represented as

So


Generally the kinetic energy at the bottom is mathematically represented as

=> 
=> 
From the law of energy conservation

=> 
=>
Answer:
1.06 secs
Explanation:
Initial speed of sled, u = 8.4 m/s
Final speed of sled, v = 5.8 m/s
Coefficient of kinetic friction, μ = 0.25
Using the impulse momentum theory, we know that the impulse applied to the sled is equal to change in momentum of the sled:
FΔt = mv - mu
where m = mass of the object
Δt = time interval
F = force applied
The force applied on the sled is the frictional force, which is given as:
F = -μmg
where g = acceleration due to gravity
Therefore:
-μmgΔt = mv - mu
-μmgΔt = m(v - u)
-μgΔt = v - u
Making Δt subject of formula:
Δt = (v - u) / -μg
Δt = (5.8 - 8.4) / (-0.25 * 9.8)
Δt = -2.6/ -2.45
Δt = 1.06 secs
It took the sled 1.06 secs to travel from A to B.
A theory of matter and energy based on the quanta concept