what force constant should I use for spring

Question

In a tube of reducing diameter a coin is dropped. At certain radius it stops going down. At this point a spring is placed such that the circumference of a coin exerts some pressure on it. As a result the spring gets very slightly displaced outward. There is a sliding split in the tube. So, for what spring constant will the coin be pushed such that it goes out of the tube throught the split. What will be the force exerrted by the coin weighing a) 2.85g  b) 6g. According to the force constant and force how much the spring will be compressed.

Answer 1

(a)

Coin weight m = 2.85 g

m = 0.00285 kg

Let coin falls from height = 1 meters

Let Spring compressed distance = x meters

Spring constant = k

Initially, there is only spring potential energy stored in the spring (remaining energies are zero),

Initial Energy  E_i = K_i + U_si + U_gi

E_i = 0 + U_si + 0

E_i = (¹/₂) k x²

Later, at the top of coin "jump", when its velocity is momentarily zero (remaining energies are zero),

There is only gravitational potential energy,

Final Energy E_f = K_f + U_sf + U_gf

E_f  = 0 + 0 + U_gf

E_f = m g h_f = (0.00285 kg) (9.8 m/s²) (1) = 0.002793 J

From conservation of energy principle : E_i = E_f

(¹/₂) k x² = 0.002793

k x² = 0.002793

Spring constant = k = 0.002793/x²

Spring compressed distance = x = 0.053/√k meters

Answer 2

(b)

Coin weight m = 6 g

m = 0.006 kg

Let coin falls from height = 1 meters

Let Spring compressed distance = x meters

Spring constant = k

Initially, there is only spring potential energy stored in the spring (remaining energies are zero),

Initial Energy  E_i = K_i + U_si + U_gi

E_i = 0 + U_si + 0

E_i = (¹/₂) k x²

Later, at the top of coin "jump", when its velocity is momentarily zero (remaining energies are zero),

There is only gravitational potential energy,

Final Energy E_f = K_f + U_sf + U_gf

E_f  = 0 + 0 + U_gf

E_f = m g h_f = (0.006 kg) (9.8 m/s²) (1) = 0.0588 J

From conservation of energy principle : E_i = E_f

(¹/₂) k x² = 0.0.0588

k x² = 0.1176

Spring constant = k = 0.1176/x²

Spring compressed distance = x = 0.34/√k

Comments

Thank you very much!!!