Question

Batman, whose mass is 89.5 kg, is holding on to the free end of a 13.4 m rope, the other end of which is fixed to a tree limb above. He is able to get the rope in motion as only Batman knows how, eventually getting it to swing enough that he can reach a ledge when the rope makes a 72.4 ◦ angle with the vertical. The acceleration of gravity is 9.8 m/s 2 . How much work was done against the force of gravity in this maneuver?

Answer 1

To calculate the work done against the force of gravity, we need to find the vertical displacement and the force due to gravity. By using the sine function to find the displacement and multiplying the mass of Batman by gravity to find the force due to gravity, we can calculate the work done against the force of gravity using the appropriate formula.

Step-by-step explanation:

To calculate the work done against the force of gravity, we need to find the vertical displacement and the force due to gravity.

First, we can find the vertical displacement by using the sine function:

displacement = length of rope x sin(angle with the vertical)

displacement = 13.4 m x sin(72.4°)

displacement = 13.1 m

Next, we can calculate the force due to gravity by multiplying the mass of Batman by gravity:

force due to gravity = mass x acceleration due to gravity

force due to gravity = 89.5 kg x 9.8 m/s^2

force due to gravity = 876.1 N

Finally, we can calculate the work done against the force of gravity using the formula:

work = force x displacement x cos(angle between the force and displacement)

work = 876.1 N x 13.1 m x cos(180°)

work = - 11,477.91 J

Answer 2

`8.2* 10^(3)` J

Step-by-step explanation:

`L` = length of the rope connected = 13.4 m

`\theta` = Angle made by the rope with the vertical = 72.4°

`h` = height gained by the batman above the reference point

Height gained by the batman above the reference point is given as

`h = L(1 - Cos\theta )`

`m` = mass of batman = 89.5 kg

`g` = acceleration due to gravity = 9.8 ms⁻²

Work done against gravity is same as the gravitational potential energy gained by batman and is given as

`W = mgh`

`W = mgL(1 - Cos\theta )`

Inserting the above values

`W = (89.5)(9.8)(13.4)(1 - Cos72.4 )`

`W = 8.2* 10^(3)` J