A small, spherical bead of mass 1.9 g is released from rest at t = 0 in a bottle of liquid shampoo. The differential equation of motion

is dv/dt = g−b/m*v .
The terminal speed is observed to be 1.6 cm/s.
The acceleration of gravity is 9.8 m/s2 .
Find the value of the constant b in the
integral form of the above equation
v =mg/b*(�1 − e^−b t/m�).
Answer in units of Ns/m.

To find the value of the constant b, we will use the given information about the terminal speed and the equation for velocity in terms of b.

Step 1: Set up the equation for terminal speed
The terminal speed is observed to be 1.6 cm/s. We need to convert it to m/s.
1 cm = 0.01 m, so 1.6 cm/s = 0.016 m/s.

At terminal speed, the net force on the bead is zero, so the acceleration is zero.
Therefore, we can set dv/dt = 0 in the differential equation:
0 = g - (b/m) * v_terminal

Step 2: Substitute the value of terminal speed
Plug in the value of v_terminal into the equation:
0 = g - (b/m) * 0.016

Step 3: Solve for b
Rearrange the equation to solve for b:
(b/m) * 0.016 = g
b/m = g / 0.016
b = (m * g) / 0.016
Now we need to substitute the given values to find the value of b.

Step 4: Substitute the given values and calculate b
The mass of the bead is given as 1.9 g. We need to convert it to kg.
1 g = 0.001 kg, so 1.9 g = 0.0019 kg.

Substitute the values into the equation:
b = (0.0019 kg * 9.8 m/s^2) / 0.016

Calculating this expression gives us the value of b in units of Ns/m.