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Answer :
Final answer:
To increase the speed by a factor of 1.70, you must increase the exerted force by a factor of 2.89, following the principle that force required for circular motion is proportional to the square of the speed of the object over its radius.
Explanation:
The question asks, "By what factor must you increase the force you exert on the rope to cause the speed to increase by a factor of 1.70?" According to the dynamics of circular motion, when considering the relationship between the force needed to maintain an object in circular motion and the speed of that object, we know that the centripetal force required for circular motion is proportional to the square of the speed of the object over its radius (F ∼ v2/r). Therefore, if you want to increase the speed of the object by a factor of 1.70, you need to increase the force by the square of that factor (1.702). The computation would be 1.702 = 2.89. This means you must increase the force by a factor of 2.89 to achieve a 1.70 times increase in speed while maintaining the same radius of motion.
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Rewritten by : Jeany
1.30 increases the force you exert on the rope to cause the speed to increase by a factor of 1.70.
What is force?
Force is a push or pull upon an object resulting from the object's interaction with another object. Every force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.
The relationship between the force applied to a rope, its mass, and the resulting acceleration is described by Newton's second law of motion.
F = m * a.
If we have an initial force F1 that causes an object of mass m to have an initial acceleration a1, and we increase the force to F2, the new acceleration will be a2. We can use this equation to find the factor by which we need to increase the force in order to cause the speed to increase by a factor of 1.70.
F2 = m * a2
F2 = m * a1 * √(1.70)
F2 = F1 * √(1.70)
So, to increase the speed by a factor of 1.70, you must increase the force you exert on the rope by the square root of 1.70, or about 1.30.
Therefore, 1.30 increases the force you exert on the rope.
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