Question

Starting from rest, a 36.7-gram steel ball sinks into a vat of corn syrup. The thick syrup exerts a viscous drag force that is proportional to the ball\'s velocity:

Drag= -Cv

where C = 0.270 N·s/m is a constant related to the size and composition of the ball as well as the viscosity of the syrup. Find the rate at which gravitational energy is converted to thermal energy once the ball reaches terminal velocity. (in W)

Answer 1

Solution:

Let us go to the basics first.

Once the terminal velocity Vt of the body is reached, it begins to move downward at constant speed. The drag force Fd then attains a constant value:

Fd = - C • Vt

At this value of Fd the rate of conversion of gravitational potential energy to thermal energy becomes equal to the power dissipated, P by the drag force at the terminal velocity:

P = Fd • Vt = - C • Vt ²

To get the terminal velocity, solve the equation of motion:

my′′ = mv′ = - mg ‒ C v

---> v′ = - g ‒ [ C / m ] v = - g ‒ γ v

(where γ = C / m)

so that v′ = d v / d t = - γ [ ( g / γ ) + v ]

Now, separating the variables, we get:

dv / [ ( g / γ ) + v ] = - γ dt which can be easily integrated to give:

ln [ ( g / γ ) + v ] = - γ t + A

---> ( g / γ ) + v = exp [ - γ t + A ] = A exp [ - γ t ]

Thus, v ( t ) = - ( g / γ ) + A exp [ - γ t ] where A = integration constant.

By definition of terminal velocity

Vt = v ( ∞ ) = - ( g / γ ) = - mg / C (negative since the body moves downward along the negative y-axis.

It follows that power P will be given by:
P = - C • Vt ² = - C • ( - mg / C ) ² = - ( mg ) ² / C

= - [ ( 0.0367 kg ) ( 9.8 m / s ² ) ] ² / ( 0.270 N • s / m)

= - 0.48 W

Taking positive value for power, thus

P = 0.48 W (Answer)

Once the ball reaches terminal velocity, gravitational potential energy is converted to thermal energy at the rate of (0.48 W)

Thanks!!!