Mid-Chapter Quiz
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Use the inverse trigonometric functions. The inverse function of sine is Arcsine.
Substitute the measure of the angle found in Part A into the equation tan θ = v^2gR.
Use the inverse tangent to find the measure of the angle.
θ ≈ 11.5^(∘)
4 meters per second
θ ≈ 9.4^(∘)
We are told that a child is seated on an outside horse of a merry-go-round that has a diameter of 16 meters. When the merry-go-round moves, the child is inclined in relation to the vertical axis with an angle θ.
Knowing that the sine of the angle of inclination is 15, we can find its measure.
In Part A we found that the angle of inclination of the child θ is equal to approximately 11.5 ^(∘). Now, to find the velocity of the merry-go-round, we will use the equation that we have been given.
tan θ = v^2/gR
Substitute values
Calculate root
Round to nearest integer
The velocity of the merry-go-round is approximately 4 meters per second.
Let's look once again at the equation for θ. We want to know the angle of inclination of the rider, if the speed of the merry-go-round is 3.6 meters per second.
tan θ = v^2/gR
Recall that in the equation, v is the velocity, R is the radius of the circular path, and g is the acceleration of gravity — which measures 9.8 meters per second. We found in Part B that the radius of the circle is 8 meters. Now we can substitute the known measures in the equation and solve for the tan θ.
Substitute values
Round to 4 decimal place(s)
To solve for θ, we will use the inverse tangent. tan θ = 0.1653 ⇒ θ = Tan^(- 1) (0.1653) We can use a calculator to find the value of θ.
Use a calculator
Round to 1 decimal place(s)
We found that θ measures approximately 9.4^(∘).