Answer:
Part a)

Part b)

Explanation:
Part A)
As we know that time period of the motion is given as

so we have



now at the point of maximum amplitude the force equation when Normal force is about to zero is given as

so we have



Part b)
Now if the amplitude of the SHM is 6.23 cm
and now at this amplitude if object will lose the contact then in that case again we have



so now we have


Answer:
14.2 m/s
Explanation:
Given data:
Speed of the stream, v₁ = 7.1 m/s
let the cross section area at initial point be A₁
now area at the second point, A₂ = (1/2)A₁ = 0.5A₁
now, from the continuity equation, we have
A₁v₁ = A₂v₂
where, v₂ is the velocity at the narrowed portion
thus, on substituting the values, we get
A₁ × 7.1 = 0.5A₁ × v₂
or
v₂ = 14.2 m/s
Given that,
Initial velocity , Vi = 0
Final velocity , Vf = 40 m/s
Acceleration due to gravity , a = 9.81 m/s²
Distance can be calculated as,
2as = Vf² - Vi²
2 * 9.81 *s = 40² - 0²
s = 81.55 m
For half height, that is, s = 40.77m
Vf= ??
2as = Vf² - Vi²
2 * 9.81 * 40.77 = Vf² - 0²
Vf² = 800
Vf = 28.28 m/s
Therefore, speed of roller coaster when height is half of its starting point will be 28 m/s.