A ball is tied to an elastic string of length 8.0 m and swung in a horizontal circle with a velocity of 0.8 m/s. When a metallic object is tied to a rope of length 2.75 m and swung in a horizontal circle, it makes one revolution in 2.9 s. The ratio of the centripetal force in the string to the centripetal force in the rope is 1/3.0. Find the mass of the metallic object attached to the rope, if the centripetal force in the string is 0.20 N.
Centripetal force on string, F1
= m1v²/r1
F1=0.2 N
Centripetal force on rope, F2
= m2r2ω²
Since 3F1=F2, therefore F2=0.6N
Equate F2=m2r2ω²
Since all variables are known except m2,
you can readily solve for m2.
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To solve this problem, we'll use the formula for centripetal force:
F = (mv^2) / r
where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular motion.
For the ball tied to the elastic string:
F_string = (m_string * v_string^2) / r_string
And for the metallic object tied to the rope:
F_rope = (m_rope * v_rope^2) / r_rope
We are given the following information:
- The length of the elastic string is 8.0 m
- The velocity of the ball is 0.8 m/s
- The length of the rope is 2.75 m
- The time for one revolution of the metallic object is 2.9 s
- The ratio of the centripetal force in the string to the centripetal force in the rope is 1/3.0
- The centripetal force in the string is 0.20 N
Let's start by finding the centripetal force for the metallic object tied to the rope. We know that one revolution is completed in 2.9 seconds, so we can determine the velocity of the object:
v_rope = (2 * π * r_rope) / t
Plugging in the values:
v_rope = (2 * π * 2.75) / 2.9
v_rope ≈ 5.724 m/s
Now we can use the given ratio to find the centripetal force in the rope:
F_string / F_rope = 1 / 3.0
Let's denote the mass of the metallic object as m_rope.
Substituting the formulas for centripetal force and simplifying the equation:
(m_string * v_string^2) / r_string / (m_rope * v_rope^2) / r_rope = 1 / 3.0
(m_string * v_string^2 * r_rope) / (r_string * v_rope^2) = 1 / 3.0
Now we can substitute the known values and solve for m_rope:
(0.20 * 0.8^2 * 2.75) / (8.0 * 5.724^2) = 1 / 3.0
Simplifying:
(0.20 * 0.64 * 2.75) / (8.0 * 32.801) = 1 / 3.0
0.0352 / 264.808 ≈ 1 / 3.0
0.000133 = 1 / 3.0
Now, to find the mass of the metallic object attached to the rope (m_rope), we can cross-multiply and solve for it:
0.000133 * 3.0 = 1
m_rope ≈ 0.000399 kg
Therefore, the mass of the metallic object attached to the rope is approximately 0.000399 kg.