## Monday, January 2, 2017

### Mechanical Properties of Fluids Problems and Solutions Three

We are dealing with the series of problems based on fluid statics and fluid dynamics. When a fluid is in the state of motion, it has potential energy,kinetic energy and also pressure energy. We know that the total energy of the system is constant. As per Bernoulli's theorem, the sum of potential energy,kinetic energy and pressure energy per unit mass always remains constant. We can also consider the viscous force acting opposite to the motion. It is the force that always opposes the relative motion. It depends on the area of cross section of the fluid, velocity of the fluid flow and the distance between the two layers of the fluid in the motion.

Problem

A vessel is kept on a table and filled with water. From the bottom of the vessel, a orifice is made and its location is given in the problem as shown in the diagram below. We need to measure the horizontal distance at which it is going to strike the floor.

Solution

We know that the velocity of the fluid coming out of orifice is similar to a velocity of a freely falling body and it depends on the height of the fluid above the opening. Once it is out, it is under the influence of the gravity and its horizontal distance distance can be measured with the equations of motion.There is no gravity acting on it so its velocity is uniform along horizontal direction. Further problem can be solved as shown in the diagram below.

Problem

Velocity of air flow on the upper surface of the wing of airplane is 40 meter per second and on the lower surface is 30 meter per second and its area of cross section is given to us and its mass is also given to us. We need to measure the force experienced by the wing and the problem is as shown in the diagram below.

Solution

As the wing is airplane is horizontal, there is no change in potential energy and we can remove that components of Bernoulli's theorem. Applying kinetic energy data, we can get the difference in the pressure at both the cases. We know that pressure is the force per unit area and hence we can measure the force acting on the system as shown in the diagram below.

Problem

Specific gravity of two liquids combined together when they have same mass and volume is given to us and we need to measure the density of each liquid. The problem is as shown in the diagram below.

Solution

We know that the density is the ratio of mass to the volume. We can find the effective density of the system when they have equal volume and equal mass can be found as shown in the diagram below. We have all ready derived equations for both of them and it can be done as shown below.

Problem

A sphere has known density and it is falling through a fluid of known density and we need to measure the acceleration of the body and the problem is as shown in the diagram below.

Solution

When a body is moving in a fluid, its weight acts in the downward direction and the upthrust acts in the upward direction. We need not consider viscous force as there is no data about coefficient of viscosity. We can write equation for the resultant force as the difference between weight and upthrust. We can use Newton’s second law and find as shown in the diagram below.

Problem

When a polar bear jumps on to ice block it just sinks and we need to measure the weights of that ice block and the specific gravity of ice and sea water is given to us. Problem is as shown in the diagram below.

Solution

We know that weight of the liquid displaced is the sum of weights of ice block and polar bear. We can equate the data and solve the problem as shown in the diagram below.

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