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Generalized mean (power mean)

In mathematics, generalised means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and ... more

Worksheet 306

Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in the figure below, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

(a) The figure shows the forearm of a person holding a book. The biceps exert a force FB to support the weight of the forearm and the book. The triceps are assumed to be relaxed. (b) Here, you can view an approximately equivalent mechanical system with the pivot at the elbow joint


There are four forces acting on the forearm and its load (the system of interest). The magnitude of the force of the biceps is FB, that of the elbow joint is FE, that of the weights of the forearm is wa , and its load is wb. Two of these are unknown FB, so that the first condition for equilibrium cannot by itself yield FB . But if we use the second condition and choose the pivot to be at the elbow, then the torque due to FE is zero, and the only unknown becomes FB .


The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the second condition for equilibrium (net τ = 0) becomes

Force (Newton's second law)
Force (Newton's second law)

Note that sin θ = 1 for all forces, since θ = 90º for all forces. This equation can easily be solved for FB in terms of known quantities,yielding. Entering the known values gives

Mechanical equilibrium - 3=3 Torque example

which yields


Now, the combined weight of the arm and its load is known, so that the ratio of the force exerted by the biceps to the total weight is



This means that the biceps muscle is exerting a force 7.38 times the weight supported.

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.
Creative Commons License : http://creativecommons.org/licenses/by/3.0/

Worksheet 341

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.

Potential energy

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.

Relation between the mean of logarithmized sample values and the mean of non-logarithmized sample values

a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. A random ... more

Median ( for odd number of observations)

In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability ... more


The variance is a parameter that describes, in part, either the actual probability distribution of an observed population of numbers, or the theoretical ... more

Monatomic ideal gas heat capacity at constant pressure

Heat capacity or thermal capacity is a physical quantity equal to the ratio of the heat that is added to (or removed from) an object to the resulting ... more

Monatomic ideal gas heat capacity at constant volume

Heat capacity or thermal capacity is a physical quantity equal to the ratio of the heat that is added to (or removed from) an object to the resulting ... more

Specific Impulse by weight - with mass flow rate

Specific impulse (usually abbreviated Isp) is a measure of the efficiency of rocket and jet engines. By definition, it is the impulse delivered per unit of ... more

Birch's Law

Birch’s law establishes a linear relation of the compressional wave velocity of rocks and minerals of a constant average atomic weight with density ... more

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