Answer:
a. 1.642 Ns/m b. 0.0294 N c. 5 × 10⁵ ms
Explanation:
a. Presuming the resistive force R = −bv, what is the value of b (N s/m)?
Using the equation of motion on the object,
W + R = ma where W = weight of the mass = mg where m =mass of marble = 3.00 g = 0.003 kg and g = acceleration due to gravity = 9.8 m/s² , R = resistive force = -bv where v = velocity and a = acceleration of marble.
So,
mg - bv = ma
At terminal speed, a = 0,
So, mg - bvt =m(0)
mg - bvt = 0
mg = bvt
b = mg/vt since terminal speed vt = 1.79 cm/s = 0.0179 m/s
So, b = 0.003 kg × 9.8 m/s²/0.0179 m/s
b = 0.0294 kgm/s² ÷ 0.0179 m/s
b = 1.642 Ns/m
b. What is the strength of the resistive force (N) when the marble reaches terminal speed?
Since the resistive force R = -bv, at terminal speed, vt
R = -bvt
R = -1.642 Ns/m × 0.0179 m/s
R = -0.0294 N
So, its strength is 0.0294 N
(c) How long in milliseconds does it take for it to reach a speed of 0.600vt?
Using mg - bv = ma where a = dv/dt,
mg - bv = mdv/dt
g - bv/m = dv/dt
separating the variables, we have
dv/(g - bv/m) = dt
Integrating, we have
∫dv/(g - bv/m) = ∫dt
(-b/m)/(-b/m) × ∫dv/ (g - bv/m) = ∫dt
1/(-b/m) ∫(-b/m)dv/(g - bv/m) = ∫dt
1/(-b/m) ㏑(g - bv/m) = t + C
㏑(g - bv/m) = -m/bt - mC/b
㏑(g - bv/m) = -m/bt + C' (C' = -mC/b)
taking antilogarithm of both sides, we have
g - bv/m = exp(-m/bt + C')
g - bv/m = exp(-m/bt)expC'
g - bv/m = Aexp(-m/bt) (A = expC')
bv/m = g - Aexp(-m/bt)
v = mg/b - (Am/b)exp(-m/bt)
when t = 0, v = 0 (since the marble starts from rest)
0 = mg/b - (Am/b)exp(-m/b(0))
0 = mg/b - (Am/b)exp(0))
-mgb = -Am/b
A = g
v = mg/b - (mg/b)exp(-m/bt)
when v = 0.600vt = 0.600 × 0.0179 m/s = 0.01074 m/s
mg/b = 0.003 kg × 9.8 m/s²/1.642 Ns/m = 0.0179 m/s and m/b = 0.003 kg/1.642 Ns/m = 0.00183/s
So,v = mg/b - (mg/b)exp(-m/bt)
0.01074 m/s = 0.0179 m/s - (0.0179 m/s)exp[(-0.00183/s)t]
0.01074 m/s - 0.0179 m/s = - (0.0179 m/s)exp[(-0.00183/s)t]
-0.00716 m/s = - (0.0179 m/s)exp[(-0.00183/s)t]
exp[(-0.00183/s)t] = -0.00716 m/s/-0.0179 m/s
exp[(-0.00183/s)t] = 0.4
taking natural logarithm of both sides, we have
(-0.00183/s)t = ㏑(0.4)
(-0.00183/s)t = -0.9163
t = -0.9163/-0.00183
t = 500 s
t = 500 × 1000 ms
t = 5 × 100000
t = 5 × 10⁵ ms
