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A sled experiences a rocket thrust that accelerates it to the right.Each rocket creates an identical thrust **T** . As in other situations where there is only horizontal acceleration, the vertical forces cancel. The ground exerts an upward force **N** on the system that is equal in magnitude and opposite in direction to its weight,**w**.The system here is the sled, its rockets, and rider, so none of the forces between these objects are considered. The arrow representing friction ( **f** ) is drawn larger than scale.

Assumptions: The mass of the Sled remains steady throughout the operation

Strategy

Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since there is no vertical acceleration. This leaves us with only horizontal forces and a simpler one-dimensional problem. Directions are indicated with plus or minus signs, with right taken as the positive direction. See the free-body diagram in the figure.

Solution

Since acceleration, mass, and the force of friction are given, we start with Newton’s second law and look for ways to find the thrust of the engines. Since we have defined the direction of the force and acceleration as acting “to the right,” we need to consider only the magnitudes of these quantities in the calculations. Hence we begin with

**Fnet** is the net force along the horizontal direction, **m** is the rocket’s mass and **a** the acceleration.

We can see from the Figure at the top, that the engine thrusts add, while friction opposes the thrust.

**Tt** is the total thrust from the 4 rockets, **Fnet** the net force along the horizontal direction and **Ff** the force of friction.

Finally, since there are **4 rockets**, we calculate the thrust that each one provides:

**T** is the individual Thrust of each engine, **b** is the number of rocket engines

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.

http://openstaxcollege.org/textbooks/college-physics

Creative Commons License : http://creativecommons.org/licenses/by/3.0/

In the wheelbarrow of the following figure the load has a perpendicular lever arm of **7.50 cm**, while the hands have a perpendicular lever arm of **1.02 m**.**(a)** What upward force must you exert to support the wheelbarrow and its load if their combined mass is **45.0 kg**? **(b)** What force does the wheelbarrow exert on the ground?

**(a)** In the case of the wheelbarrow, the output force or load is between the pivot and the input force. The pivot is the wheel’s axle. Here, the output force is greater than the input force. Thus, a wheelbarrow enables you to lift much heavier loads than you could with your body alone. **(b)** In the case of the shovel, the input force is between the pivot and the load, but the input lever arm is shorter than the output lever arm. The pivot is at the handle held by the right hand. Here, the output force (supporting the shovel’s load) is less than the input force (from the hand nearest the load), because the input is exerted closer to the pivot than is the output.

Strategy

Here, we use the concept of mechanical advantage.

Discussion

An even longer handle would reduce the force needed to lift the load. The MA here is:

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.

http://openstaxcollege.org/textbooks/college-physics

Creative Commons License : http://creativecommons.org/licenses/by/3.0/

Subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then ... more

In a video game design, a map shows the location of other characters relative to the player, who is situated at the origin, and the direction they are facing. A character currently shows on the map at coordinates **(-3, 5)**. If the player rotates counterclockwise by **20 degrees**, then the objects in the map will correspondingly rotate **20 degrees** clockwise. Find the new coordinates of the character.

To rotate the position of the character, we can imagine it as a point on a circle, and we will change the angle of the point by **20 degrees**. To do so, we first need to find the radius of this circle and the original angle.

Drawing a right triangle inside the circle, we can find the radius using the Pythagorean Theorem:

To find the angle, we need to decide first if we are going to find the acute angle of the triangle, the reference angle, or if we are going to find the angle measured in standard position. While either approach will work, in this case we will do the latter. By applying the cosine function and using our given information we get

While there are two angles that have this cosine value, the angle of **120.964** degrees is in the second quadrant as desired, so it is the angle we were looking for.

Rotating the point clockwise by **20 degrees**, the angle of the point will decrease to **100.964 degrees**. We can then evaluate the coordinates of the rotated point

For **x** axis:

For **y** axis:

The coordinates of the character on the rotated map will be **(-1.109, 5.725)**

Reference : PreCalculus: An Investigation of Functions,Edition 1.4 © 2014 David Lippman and Melonie Rasmussen

http://www.opentextbookstore.com/precalc/

Creative Commons License : http://creativecommons.org/licenses/by-sa/3.0/us/

In materials science fatigue occurs when a material is subjected to repeated loading and unloading. The failure of the material occurs when there are k ... more

The Rand index or Rand measure (named after William M. Rand) in statistics, and in particular in data clustering, is a measure of the similarity between ... more

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability ... more

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

The mass number (A), also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an ... more

The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the ... more

In fluid mechanics, the Reynolds number (Re) is a dimensionless quantity that is used to help predict similar flow patterns in different fluid flow ... more

In fluid dynamics, the Ursell number indicates the nonlinearity of long surface gravity waves on a fluid layer. This dimensionless parameter is named after ... more

Sensible heat is heat exchanged by a body or thermodynamic system that changes the temperature, and some macroscopic variables of the body, but leaves ... more

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or ... more

Calculate the Reynolds number **N′R** for a ball with a **7.40-cm** diameter thrown at **40.0 m/s**.

Strategy

We can use the Reynolds number equation calculate N’_{R} , since all values in it are either given or can be found in tables of density and viscosity.

Solution

We first find the kinematic viscosity values:

Substituting values into the equation for N’R yields:

Discussion

This value is sufficiently high to imply a turbulent wake. Most large objects, such as airplanes and sailboats, create significant turbulence as they move. As noted before, the Bernoulli principle gives only qualitatively-correct results in such situations.

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.

http://openstaxcollege.org/textbooks/college-physics

Creative Commons License : http://creativecommons.org/licenses/by/3.0/

Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. The notion of permutation relates to the act of ... more

In heat transfer at a boundary (surface) within a fluid, the Nusselt number (Nu) is the ratio of convective to conductive heat transfer across (normal to) ... more

In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the ... more

The Rayleigh–Taylor instability, or RT instability , is an instability of an interface between two fluids of different densities which occurs when the ... more

Arithmetic mean is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an ... more

The Froude number (Fr) is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be ... more

The Atwood number is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a ... more

The awe‐inspiring Great Pyramid of Cheops was built more than 4500 years ago. Its square base, originally 230 m on a side, covered 13.1 acres, and it was 146 m high (H), with a mass of about 7×10^9 kg. (The pyramid’s dimensions are slightly different today due to quarrying and some sagging). Historians estimate that 20,000 workers spent 20 years to construct it, working 12-hour days, 330 days per year.

a) Calculate the gravitational potential energy stored in the pyramid, given its center of mass is at one-fourth its height.

b) Only a fraction of the workers lifted blocks; most were involved in support services such as building ramps, bringing food and water, and hauling blocks to the site. Calculate the efficiency of the workers who did the lifting, assuming there were 1000 of them and they consumed food energy at the rate of 300 Kcal/hour.

first we calculate the number of hours worked per year.

then we calculate the number of hours worked in the 20 years.

Then we calculate the energy consumed in 20 years knowing the energy consumed per hour and the total hours worked in 20 years.

The efficiency is the resulting potential energy divided by the consumed energy.

The Weber number (We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two ... more

The Péclet number is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. It is named after the French physicist ... more

The Péclet number (Pe) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. It is named after the French ... more

In compressible fluid dynamics, impact pressure (dynamic pressure) is the difference between total pressure (also known as pitot pressure or stagnation ... more

Amagat’s law or the Law of Partial Volumes of 1880 describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) ... more

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Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust

T, for the four-rocket propulsion system shown in the Figure below. The sled’s initial acceleration is49 m/s, the mass of the system is^{2}2100 kg, and the force of friction opposing the motion is known to be650 N.