A row of evenly spaced dominoes, each with a mass = 100 g ( 0.1 kg ) and height h = 0.2 m, is placed on a flat horizontal surface. The 1rst domino that falls delivers 5J of energy to the 2nd domino in line. Energy losses due to friction and air resistance occur between each fall, with a constant energy transfer efficiency ratio of r = 0.9.
The chain reaction comes to a halt when the energy transferred to a domino is less than the energy required to tip it over ( denoted herein as “ Ef “ ). To determine when the chain reaction stops, please use the following assumption: the energy required to tip a domino is the potential energy ( Ef ) needed to lift its center of mass by a vertical distance of ½ h. Do not take thickness or rotational effects into account.
Q: How many dominoes will fall prior to the chain reaction coming to a halt???
Basic Logarithmic Function Derivation:
A: A logarithm is a mathematical operation that reveals what exponent is needed to raise some base ( 2, 10, e, etc. ) to be equal to an output value that is known. 2^1 = 2, so the exponent needed to raise a base of “ 2 “ so that it equals the number “ 2 “ is “ 1 “. Likewise, since 2^2 = 4, the exponent needed to raise base-2 to equal 4 is “ 2 “. The notation ( regardless of base usage ) for logarithms is as follows:
Ex. log(2) y = x, so 2^x = y
A more simplified base ( e ) comes from calculus, and its notation is as follows:
Ex. ln(e) y ( simplified to ln y ) = x, so e^x = y
Things become a lot trickier in proportion to the value of ( y ) that enables ln y = x to be a true statement. Keeping in mind that a logarithm is an exponent, consider the following cases where e^x may equal a product, quotient, or value that itself is being raised by some exponential value:
Ex. e^x = y, so ln y = x
If ln y = x, then e^x = y must also be expressed as e^ln y = y
And conversely,
lf e^ln y = y, then by default, ln e^ln y = y
ln y = ( some exponent ) that raises base e^( some exponent ) = y, so e^( some exponent ) = y, then e^ln y = y.
Logarithmic functions are inverse operations of exponential functions.
In the same way that division is the inverse of multiplication, and multiplication is the inverse of division, e^x = y is the inverse operation of log y = x.
Repeating for clarity:
If e^ln y = y, let’s first evaluate the ln y expression separately, realizing that its value is an exponent. The ln y exponent represents an exponent that raises the base e to equal “ y “, so e^( some exponent that raises e to be equal to y ) e^ln y = y
Time spent reviewing this topic until an intuitive understanding is developed cannot be overstated!!!
The important thing to keep in mind is that a logarithm is an alternative mathematical expression for the value of an exponent, and as such, the output of a logarithmic function must be reflective of the rules of the exponents that apply to quantities that are multiplied, divided, or even and output value that unto itself is being raised by an exponent.
Conceptual derivation of the answer:
Ef will be used to represent the energy barrier that a single domino cannot pass after ( n ) number of falls have occurred due to sequential energy losses that occur with each fall. Our answer will necessitate converting 100 g into its kilogram equivalent:
Energy in joules ( J ) = Force ( N ) x distance ( m ) = N*m, where F = ma = mg ( g = gravitational constant of acceleration of 9.8 m/s^2 ).
Ef = 1/2mgh = ( 0.5 )( 0.1 kg )( 9.8 m/s^2 )( 0.2 m ) = 0.098 J
The problem states that the 2nd domino in the chain reaction receives 5J of energy from the 1rst, so the 3rd domino is the one that will receive some fraction of 5J due to energy that the 2nd domino loses to air friction; therefore, the 3rd domino in the sequence gains ( 5 J )( 0.9 ) joules of energy, the fourth domino has ( 5 J )( 0.9 )( 0.9 ) joules passed to it, because it gains a fraction of the fraction of energy passes to the domino before it ( 90% of 90% ). Mathematically, the 3rd domino’s energy is equal to E = ( 5 J )( 0.9 )^1, and the 4th domino’s energy is E = ( 5 J )( 0.9 )^2. This shows that whatever number ( n ) of dominoes that have fallen, the exponent that ( 0.9 ) is raised to will be two digits smaller ( n – 2 ).
A general equation for the energy of the nth domino to fall is as follows:
E = ( Eo )( 0.9 )^n-2
Rules of logarithms:
An interesting rule of logarithms will be used to determine what value of n will give us an answer. A logarithm answers the question of what exponent is needed to raise some base so that it is equal to some given value at hand. Let’s look at an example where the base in question is the number ” 10 “, and we want to know what exponent is needed to raise the base 10 so that it is equal to 10,000. It is understood that a logarithm expressed as a ” log ” refers to a base of 10, so the base is omitted from the express we use:
log 100^2 = y
log 100^2 = some exponent that will raise a base of 10 to equal the number 100 that is itself being raised by the exponent ” 2 “.
10^y = 100
10^2 = 100
10^( 2 )^2 = ???
10^( 2 )( 2 ) = 10^4 = 10,000
Likewise,
log 100^2 = ???
( 2 )( log 100 ) = ???
( 2 )( 2 ) = 2^2 = 4
10^4 = 10,000
We now have all of the mathematical and conceptual tools needed to answer the question at hand.
10^10 = ( 100 ), so in accordance with the rule regarding exponents that themselves have exponents, the exponent needed to raise a base of 10 to 100 must in turn be multiplied by this output in order to reflect the value we are looking for. This is why the rule of logs allows for the following mathematical maneuver:
The last domino that will fall will have an energy level that is either equal or close to being equal to the tipping energy ( Ef ) derived earlier ( 0.098 J ).
Due to the fact that a logarithm with a base of ” e “, which is an irrational number, makes calculus so much simpler, exponential problems that use log functions to derive solutions oftentimes use the natural log of e ( ln e ) to solve problems, which shall be the case here with ln e = ln ):
Note: I don’t have a neat way of expressing ” greater than or equal to “, so I’m going to use the ” equal to ” expression to solve the problem at hand.
( Eo )( r )^n-2 = Ef
( r )^n-2 = ( Ef / Eo )
ln ( r )^n-2 = ln ( Ef / Eo )
The step above used to ensure that our derived exponent is multiplied by the value that our base is being raised to is used once again:
( n – 2 )( ln r ) = ln ( Ef / Eo )
n – 2 = ln ( Ef / Eo ) ] / ln r
Note: When dealing with inequalities, we must take care to invert the inequality when division with a negative number occurs ( as is the case here, because the natural log of a fraction ( r = 0.9 ) yields a negative value.
n – 2 = ln ( 0.0196 ) / ( ln 0.9 )
n – 2 = ( – 3.932225713 / – 0.105360516 )
n – 2 = 37.32162543
n = 39.32162543
39 dominoes will drop in the chain reaction and then come to a halt.
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