) A physics student wants to measure the stiffness of a spring (force required per cm stretched). He knows that according to Hoo
ke's law, there is a linear relationship between the distance a spring is stretched and the force needed to stretch the spring. He collects some data by measuring the force applied to the spring when he stretches the spring by some amount. The plot and the least squares fit is given below. From the regression model, the intercept was found to be -2.532 and the slope was found to be 25.321. Part i). The stiffness of the spring was predicted to be A. 125.84 B. -2.532 C. -9.961 D. 25.321 E. 50.642 Part ii). Refer to the previous question, the physics student used the regression model to predict that a force of 377.28N would be required to stretch the spring by 15cm. Remarkably, his prediction was horribly wrong. Can you explain why
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The charge of the copper nucleus is 29 times the charge of one proton:
the charge of the electron is
and their separation is
The magnitude of the electrostatic force between them is given by:
where
is the Coulomb's constant. If we substitute the numbers, we find (we can ignore the negative sign of the electron charge, since we are interested only in the magnitude of the force)
the charge of the electron is
and their separation is
The magnitude of the electrostatic force between them is given by:
where
Answer:
The frequency = 521.59 Hz
The rate at which the frequency is changing = 186.9 Hz/s
Explanation:
Given that :
Diameter of the tank = 44 cm
Radius of the tank = =
= 22 cm
Diameter of the spigot = 3.0 mm
Radius of the spigot = =
= 1.5 mm
Diameter of the cylinder = 2.0 cm
Radius of the cylinder = =
= 1.0 cm
Height of the cylinder = 40 cm = 0.40 m
The height of the water in the tank from the spigot = 35 cm = 0.35 m
Velocity at the top of the tank = 0 m/s
From the question given, we need to consider that the question talks about movement of fluid through an open-closed pipe; as such it obeys Bernoulli's Equation and the constant discharge condition.
The expression for Bernoulli's Equation is as follows:
where;
P₁ and P₂ = initial and final pressure.
v₁ and v₂ = initial and final fluid velocity
y₁ and y₂ = initial and final height
p = density
g = acceleration due to gravity
So, from our given parameters; let's replace
v₁ = 0 m/s ; y₁ = 0.35 m ; y₂ = 0 m ; g = 9.8 m/s²
∴ we have:
v₂ =
v₂ =
v₂ = 2.61916
v₂ ≅ 2.62 m/s
Similarly, using the expression of the continuity for water flowing through the spigot into the cylinder; we have:
v₂A₂ = v₃A₃
v₂r₂² = v₃r₃²
where;
v₂r₂ = velocity of the fluid and radius at the spigot
v₃r₃ = velocity of the fluid and radius at the cylinder
where;
v₂ = 2.62 m/s
r₂ = 1.5 mm
r₃ = 1.0 cm
we have;
v₃ =
v₃ = 0.0589 m/s
∴ velocity of the fluid in the cylinder = 0.0589 m/s
So, in an open-closed system we are dealing with; the frequency can be calculated by using the expression;
where;
= velocity of sound
h = height of the fluid
v₃ = velocity of the fluid in the cylinder
= 521.59 Hz
∴ The frequency = 521.59 Hz
b)
What are the rate at which the frequency is changing (Hz/s) when the cylinder has been filling for 4.0 s?
The rate at which the frequency is changing is related to the function of time (t) and as such:
where;
(velocity of sound) = 343 m/s
v₃ (velocity of the fluid in the cylinder) = 0.0589 m/s
h (height of the cylinder) = 0.40 m
t (time) = 4.0 s
Substituting our values; we have ;
= 186.873
≅ 186.9 Hz/s
∴ The rate at which the frequency is changing = 186.9 Hz/s when the cylinder has been filling for 4.0 s.