A ball has a diameter of 60 mm and falls in oil with a terminal velocity of 0.8 m / s. Determi

step 1 At terminal velocity, weight minus the buoyancy is equal to the drag. Now we're given D is 0 .06

A ball has a diameter of 60 mm and falls in oil with a...

A ball has a diameter of $60 \mathrm{~mm}$ and falls in oil with a terminal velocity of $0.8 \mathrm{~m} / \mathrm{s}$. Determine the density of the ball. For oil, take $\rho_{o}=880 \mathrm{~kg} / \mathrm{m}^{3}$ and $\nu_{0}=40\left(10^{-6}\right) \mathrm{m}^{2} / \mathrm{s}$.

Key Concepts

Reynolds Number

The Reynolds number is a dimensionless quantity that characterizes the flow regime around an object moving in a fluid. It is a measure of the ratio of inertial forces to viscous forces. Determining the Reynolds number helps in selecting the appropriate drag model, as different flow regimes (laminar or turbulent) have different drag characteristics.

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. It is determined by the weight of the fluid displaced by the object. In the context of an object falling in a fluid, the buoyant force decreases the effective weight and is a key factor in calculating the net force acting on the object.

Drag Coefficient

The drag coefficient is a dimensionless factor that quantifies the drag or resistance of an object in a fluid environment. It encapsulates the effects of shape, flow conditions, and surface roughness, and is used in the drag force equation to relate the fluid properties and object geometry to the resistive force encountered at terminal velocity.

Force Balance at Terminal Velocity

At terminal velocity, a falling object experiences a balance of forces: the gravitational force acting downward is counteracted by both the buoyant force (upward thrust due to displaced fluid) and the drag force (resistance from the fluid). This equilibrium condition enables us to derive relationships among the object's properties (like density and size) and the fluid properties.

Terminal Velocity

Terminal velocity is reached when an object falling through a fluid attains a constant speed because the sum of the forces acting on it (gravity, buoyancy, and drag) becomes zero. This principle is crucial because it allows us to set up an equation where the driving force (net weight) is exactly balanced by the resistive force (drag), simplifying the analysis of the motion.

Drag Force

Drag force is the resistive force experienced by an object moving through a fluid. It depends on factors such as the object's shape, cross-sectional area, the fluid's density, and the square of the object's velocity. Understanding drag force is essential to solve problems involving terminal velocity because it quantifies the fluid's resistance to the object's motion.

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