Batman, whose mass is 119 kg, is holding on
to the free end of a 14.6 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 52.4� angle with the vertical.
The acceleration of gravity is 9.8 m/s2 .
How much work was done against the force
of gravity in this maneuver?
Answer in units of J.
It didn’t work ho
To calculate the work done against the force of gravity in this maneuver, we can use the formula for work:
Work = force x displacement x cos(theta)
In this case, the force we need to consider is the weight of Batman, which is given by the formula:
Weight = mass x gravity
where the mass of Batman is 119 kg and the acceleration due to gravity is 9.8 m/s^2.
First, let's calculate the weight:
Weight = 119 kg x 9.8 m/s^2
Weight = 1166.2 N
Next, let's calculate the displacement. Since Batman swings in a circular motion, the displacement is the distance traveled along the arc of the swing. This can be calculated using the formula:
Displacement = circumference x (angle / 360)
where the circumference of the swing is given by the formula:
Circumference = 2 x pi x radius
Since the length of the rope is given as 14.6 m, the radius of the swing is half of that:
Radius = 14.6 m / 2
Radius = 7.3 m
Now, let's calculate the circumference:
Circumference = 2 x pi x 7.3 m
Circumference = 45.8 m
Now, let's calculate the displacement:
Displacement = 45.8 m x (52.4 / 360)
Displacement = 6.66 m
Finally, let's calculate the work:
Work = 1166.2 N x 6.66 m x cos(0°)
Work = 1166.2 N x 6.66 m x 1
Work = 7770.01 J
Therefore, the work done against the force of gravity in this maneuver is approximately 7770.01 Joules.
To find the work done against the force of gravity in this maneuver, we can use the formula:
Work = force × displacement × cosθ
The force we need to consider is the component of the force of gravity acting along the direction of the displacement. This component can be calculated using the mass and acceleration due to gravity:
Force of gravity = mass × acceleration due to gravity
Now, let's break down the solution step by step:
Step 1: Calculate the force of gravity
Force of gravity = mass × acceleration due to gravity
Force of gravity = 119 kg × 9.8 m/s²
Force of gravity = 1166.2 N
Step 2: Calculate the displacement (vertical height)
The displacement is given by the length of the rope and the angle it makes with the vertical. We can use trigonometry to find the vertical height.
Vertical height = length of the rope × sinθ
Vertical height = 14.6 m × sin(52.4º)
Vertical height = 11.44 m
Step 3: Calculate the work done against gravity
Work = force × displacement × cosθ
Work = 1166.2 N × 11.44 m × cos(52.4º)
Now, let's calculate the final result:
Work = 1166.2 N × 11.44 m × cos(52.4º)
Work ≈ 6,421.4 J
Therefore, the amount of work done against the force of gravity in this maneuver is approximately 6,421.4 J (joules).
m g h
where h = 14.6 (1 - cos 52.4)
draw picture :)