A horizontal clothesline is tied between 2 poles, 12 meters apart. When a mass of 5 kilograms is tied to the middle of the clothesline, it sags a distance of 1 meters. What is the magnitude of the tension on the ends of the clothesline?
A constant force moves an object along a straight line from point (10, 2, -6) to point (10, 5, -10).
Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.
A woman exerts a horizontal force of 8 pounds on a box as she pushes it up a ramp that is 6 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box.
To find the magnitude of the tension on the ends of the clothesline, we can use the concept of equilibrium. When the clothesline is sagging, it forms a right triangle with the horizontal distance between the poles and the sagging distance as its two legs. The weight of the mass creates a vertical force that balances out the tension forces on either end of the clothesline.
To calculate the tension, we can use the Pythagorean theorem to find the length of the hypotenuse of the right triangle formed:
tension² = weight² + sagging distance²
Given that the weight is 5 kilograms and the sagging distance is 1 meter, we can substitute these values into the equation:
tension² = 5² + 1²
tension² = 26
tension = √26, approximately 5.1 Newtons
Therefore, the magnitude of the tension on the ends of the clothesline is approximately 5.1 Newtons.
To find the work done when a constant force moves an object along a straight line, we can use the formula:
work done = magnitude of force * distance
In this case, the distance between the two points is the displacement of the object. To find the displacement, we can calculate the differences between the respective coordinates:
Δx = 10 - 10 = 0 meters
Δy = 5 - 2 = 3 meters
Δz = -10 - (-6) = -4 meters
The displacement is the vector sum of these differences:
displacement = √(Δx² + Δy² + Δz²) = √(0² + 3² + (-4)²) = √(0 + 9 + 16) = √25 = 5 meters
Given that the distance is 5 meters and the magnitude of the force is measured in newtons, the work done is:
work done = magnitude of force * distance = force * distance
Therefore, the work done is equal to the force applied multiplied by the distance, but without the given force value, we cannot provide a specific answer to this question.
To find the work done on the box when a woman exerts a force on it, we can use the equation:
work done = force * distance * cos(angle)
In this case, the force applied by the woman is 8 pounds, and the distance is 6 feet. However, since the problem specifies that the angle is inclined at 30 degrees above the horizontal, we need to convert pounds and feet into standard units of force (newtons) and distance (meters) respectively.
1 pound is approximately equal to 4.448 newtons.
1 foot is approximately equal to 0.3048 meters.
Converting the units, we get:
force = 8 pounds * 4.448 newtons/pound ≈ 35.584 newtons
distance = 6 feet * 0.3048 meters/foot ≈ 1.8288 meters
Plugging the values into the equation, we have:
work done = 35.584 newtons * 1.8288 meters * cos(30 degrees)
Using the cosine of 30 degrees, which is approximately 0.866, we can calculate the work done:
work done ≈ 35.584 newtons * 1.8288 meters * 0.866
work done ≈ 55.018 Joules
Therefore, the work done on the box is approximately 55.018 Joules.