Observe that the given vector field is a gradient field:
Let 
, so that
Integrating the first equation with respect to 
, we get
Differentiating this with respect to 
gives
Now differentiating 
with respect to 
gives
Putting everything together, we find a scalar potential function whose gradient is 
,
It follows that the curl of is 0 (i.e. the zero vector).
Answer:
a) ΔL/L = F / (E A), b) 
= L (1 + L F /(EA) )
Explanation:
Let's write the formula for Young's module
E = P / (ΔL / L)
Let's rewrite the formula, to have the pressure alone
P = E ΔL / L
The pressure is defined as
P = F / A
Let's replace
F / A = E ΔL / L
F = E A ΔL / L
ΔL / L = F / (E A)
b) To calculate the elongation we must have the variation of the length, so the length of the bar must be a fact. Let's clear
ΔL = L [F / EA]
-L = L (F / EA)
= L + L (F / EA)
= L (1 + L (F / EA))
Answer:
At the top of the hill.
Explanation:
As the roller coaster goes up the hill, kinetic energy (K.E) decreases, gravitational potential energy (G.P.E) increases .
As it reach the top of the hill, K.E becomes zero and G.P.E reaches <em>m</em><em>a</em><em>x</em><em>i</em><em>m</em><em>u</em><em>m</em> .
As it goes down the hill, K.E starts to increase and G.P.E decrease .
At the bottom of the hill, K.E reaches <em>maximum</em> and G.P.E becomes zero .
(Correct me it I am wrong)
In a gear train with two gears, the gear ratio is defined as follows
where
is the angular velocity of the input gear while
is the angular velocity of the output gear.
This can be rewritten as a function of the number of teeth of the gears. In fact, the angular velocity of a gear is inversely proportional to the radius r of the gear:
But the radius is proportional to the number of teeth N of the gear. Therefore we can rewrite the gear ratio also as
