One 2.5 kg paint bucket is hanging by a massless cord from another 2.5 kg paint bucket, also hanging by a massless cord, as shown in Fig. 4-44.
(a) If the buckets are at rest, what is the tension in each cord?
(lower cord)________ N
(upper cord)________ N
(b) If the two buckets are pulled upward with an acceleration of 1.70 m/s2 by the upper cord, calculate the tension in each cord.
(lower cord)________ N
(upper cord)_________N
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(a) When the buckets are at rest, the tension in each cord can be calculated using the concept of equilibrium.
Since the buckets are at rest, the net force acting on each bucket is zero.
For the lower bucket:
The upward tension in the lower cord is balanced by the force of gravity acting downward on the lower bucket.
Therefore, the tension in the lower cord is equal to the weight of the lower bucket.
Weight = mass × acceleration due to gravity
Weight = (2.5 kg) × (9.8 m/s^2)
Tension in lower cord = 24.5 N
For the upper bucket:
The downward tension in the upper cord is balanced by the force of gravity acting downward on the upper bucket.
Therefore, the tension in the upper cord is equal to the weight of the upper bucket.
Tension in upper cord = 24.5 N
(b) When the two buckets are pulled upward with an acceleration of 1.70 m/s^2 by the upper cord, the tension in each cord can be calculated using Newton's second law (F=ma).
For the lower bucket:
The net force acting on the lower bucket is the tension in the lower cord minus the weight of the lower bucket.
Net force = Tension in lower cord - Weight = ma
Tension in lower cord = Weight + ma
Tension in lower cord = (2.5 kg) × (9.8 m/s^2) + (2.5 kg) × (1.70 m/s^2)
Tension in lower cord = 24.5 N + 4.25 N
Tension in lower cord = 28.75 N
For the upper bucket:
The net force acting on the upper bucket is the tension in the upper cord minus the weight of the upper bucket.
Net force = Tension in upper cord - Weight = ma
Tension in upper cord = Weight + ma
Tension in upper cord = (2.5 kg) × (9.8 m/s^2) + (2.5 kg) × (1.70 m/s^2)
Tension in upper cord = 24.5 N + 4.25 N
Tension in upper cord = 28.75 N
To answer these questions, we need to understand the concept of forces and tension.
(a) When the buckets are at rest, the tension in each cord is equal to the weight of the bucket it is supporting. The weight of an object can be calculated using the formula:
Weight = mass * acceleration due to gravity
In this case, the mass of each bucket is 2.5 kg. The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight of each bucket is:
Weight = 2.5 kg * 9.8 m/s^2 = 24.5 N
Since the tension in each cord is equal to the weight of the bucket it is supporting, the tension in the lower cord and upper cord is both 24.5 N.
(b) When the two buckets are pulled upward with an acceleration of 1.70 m/s^2 by the upper cord, we need to consider the net force acting on each bucket.
For the lower bucket:
The tension in the lower cord is pulling it upward, while its weight is pulling it downward. The net force on the lower bucket is given by:
Net Force = Tension in lower cord - Weight of the lower bucket
Since the weight of the lower bucket is still 24.5 N, the net force on the lower bucket is:
Net Force = Tension in lower cord - 24.5 N
According to Newton's second law (F = mass * acceleration), the net force is equal to the mass of the lower bucket multiplied by its acceleration. Therefore:
Tension in lower cord - 24.5 N = (2.5 kg) * (1.70 m/s^2)
Solving for the tension in the lower cord:
Tension in lower cord = (2.5 kg) * (1.70 m/s^2) + 24.5 N
For the upper bucket:
The tension in the upper cord is pulling it upward, while its weight is pulling it downward. The net force on the upper bucket is given by:
Net Force = Tension in upper cord - Weight of the upper bucket
Since the weight of the upper bucket is still 24.5 N, the net force on the upper bucket is:
Net Force = Tension in upper cord - 24.5 N
Using Newton's second law again:
Tension in upper cord - 24.5 N = (2.5 kg) * (1.70 m/s^2)
Solving for the tension in the upper cord:
Tension in upper cord = (2.5 kg) * (1.70 m/s^2) + 24.5 N
Note: The actual values of the tensions can be calculated by substituting the given values into the equations.
a.)1st-> 2.5*2 = 5kg
5kg(9.8m/s^2) = 49N
2nd-> 2.5kg(9.8m/s^2)= 24.5N
b.)1st-> 5kg(9.8m/s^2 - 1.70m/s^2)
= 5kg(8.1m/s^2)
= 40.5N
2nd-> 2.5kg(8.1m/s^2)
= 20.25N