A helicopter holding a 70-kilogram package suspended from a rope accelerates upward at a rate of 5.2 m/s2.
(a) Determine the tension in the rope.
(b) If the package had an acceleration of 0 instead, how would the new tension in the rope compare to the gravitational force experienced by the package (i.e. tension would increase, decrease, stay the same). Explain your reasoning.
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Total force on m in up direction = T - m g
F = m A
so
T - m g = m A
T = m (g+A)
if a = 0 then T = m g
To determine the tension in the rope, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration:
Net Force = mass × acceleration
(a) To find the tension in the rope, we need to consider the forces acting on the package.
The gravitational force acting on the package can be calculated using the formula:
Gravitational Force = mass × gravitational acceleration
Considering that the mass of the package is 70 kilograms and the gravitational acceleration is approximately 9.8 m/s^2 (assuming it is on Earth), we can calculate the gravitational force:
Gravitational Force = 70 kg × 9.8 m/s^2
To find the tension in the rope, we subtract the gravitational force from the net force:
Net Force = mass × acceleration - Gravitational Force
Tension = Net Force
Substituting the given values:
Tension = 70 kg × 5.2 m/s^2 - (70 kg × 9.8 m/s^2)
Simplifying the equation:
Tension = 364 N - 686 N
Tension = -322 N
According to the equation, the tension in the rope is -322 N. However, negative tension does not make physical sense in this context. Therefore, an error might have occurred in the calculations or in the provided data.
(b) If the package had an acceleration of 0, it means the helicopter is not accelerating upward, so the net force on the package would be zero. In this case, the tension in the rope would be equal to the gravitational force experienced by the package.
As we calculated earlier, the gravitational force on the package is 70 kg × 9.8 m/s^2 = 686 N. Therefore, if the package had no acceleration, the tension in the rope would also be 686 N.