Question

Consider a region where the gravitational field of 10 N/kg is uniform and points in the -y direction. A 2 kg object at the

origin O is moved along the path y = x² to point A (x = 2 m, y = 4 m). The object is then moved along the path (x - 2) =
(y-4)2 to the point B (x = 6 m, y = 6 m), as shown. How much work is done by gravity as the object is moved from O to
B?
o
C
d
e
B
oj
-80 J
80J
120J
-120J

Answer 1

The work done by gravity moving a 2 kg object from O to B, given a gravitational field of 10 N/kg, is -120 J. The negative sign indicates work against gravity.

The work done by gravity as an object moves in a gravitational field is given by the formula:

`\[ W = -mgh \]`

where:

-
`\( W \)` is the work done,

- m is the mass of the object,

- g is the gravitational field strength,

- h is the vertical displacement.

In this case, the gravitational field is given as
`\( 10 \, \text{N/kg} \)` in the -y direction, and the object is moved along a path in the xy-plane. The displacement vector
`\(\vec{d}\)` is parallel to the gravitational field, so the work done is given by:

`\[ W = -mgh \]`

Since the gravitational field is
`\(10 \, \text{N/kg}\)`, and the mass
`(\(m\)) is \(2 \, \text{kg}\),` we can calculate
`\(g \cdot h\)` using the vertical displacement
`(\(h\)).`

The vertical displacement
`(\(h\))` is the difference in y-coordinates between the initial and final positions of the object.

1. From O to A:

`\(h_1 = 4 \, \text{m}\) (final y-coordinate) - \(0 \, \text{m}\) (initial y-coordinate) \(= 4 \, \text{m}\).`

2. From A to B:

`\(h_2 = 6 \, \text{m}\) (final y-coordinate) - \(4 \, \text{m}\) (initial y-coordinate) \(= 2 \, \text{m}\).`

Now, calculate the total vertical displacement
`(\(h\)):`

`\[ h = h_1 + h_2 = 4 \, \text{m} + 2 \, \text{m} = 6 \, \text{m} \]`

Now, substitute the values into the formula:

`\[ W = -mgh = -(2 \, \text{kg})(10 \, \text{N/kg})(6 \, \text{m}) \]`

`\[ W = -120 \, \text{J} \]`

The negative sign indicates that the work is done against the gravitational field, which makes sense because the object is moving in the positive y-direction. Therefore, the correct answer is
`\(-120 \, \text{J}\).`