The diameter of the base of a tapered drinking cup is 6.6cm . The diameter of its mouth is 9.8cm . The path of the cup curves when you roll it on the top of a table.
How much faster does the mouth move than the base?
Express your answer using two significant figures.
An angular speed ω=v₁/R₁=v₂/R₂
v₁/v₂= R₁/R₂=9.8/6.6 =1.48
vm/vb=1.257
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To find out how much faster the mouth moves than the base, we need to compare the circumference of the mouth to the circumference of the base, as the path traced by the cup when rolled on the table is essentially the circumference.
The formula to calculate the circumference of a circle is C = πd, where C is the circumference and d is the diameter of the circle.
For the base of the tapered drinking cup:
C_base = π * d_base
For the mouth of the drinking cup:
C_mouth = π * d_mouth
To find out how much faster the mouth moves than the base, we can calculate the ratio of their circumferences.
Ratio = C_mouth / C_base = (π * d_mouth) / (π * d_base)
Since both the numerator and denominator have the same factor of π, we can cancel it out:
Ratio = d_mouth / d_base
Now we can substitute the given diameters into the equation:
Ratio = 9.8 cm / 6.6 cm
Calculating this, we find:
Ratio ≈ 1.49
Therefore, the mouth of the tapered drinking cup moves approximately 1.49 times faster than the base.