Question

Starting from rest, a 41.5-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:

where C = 0.281 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.

Answer 1

2 answers · Physics

Best Answer

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 which instant the rate of conversion of gravitational potential energy to thermal
energy becomes equal to the power dissipated by the drag force at the terminal
velocity …… P = Fd • Vt = C • Vt ² …… hence, determine the terminal velocity …
To get that 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 ] …. separating the variables,
we get …. d v / [ ( 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 ]
… 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 …
… P = C • Vt ² = C • ( - mg / C ) ² = ( mg ) ² / C
………………… = [ ( 0.0415 kg ) ( 9.8 m / s ² ) ] ² / ( 0.281N • s / m) = 0.5886 W
Once the ball reaches terminal velocity, gravitational potential energy is converted to
thermal energy at the rate of ( 0.5886W)