A 52kg teen grabs onto a rope swing. The length of rope between the teen and the pivot point above is 4.5m, and the rope makes an angle of 40o relative to vertical. Calculate his change in potential energy as he swings down and out to the point where the rope is in the vertical position.
Δh=L-Lcosα=L(1-cosα)
ΔPE=mg Δh=mgL(1-cosα)=
=52•9.8•4.5•(1-cos40°) = …
To calculate the change in potential energy, we need to find the difference in height between the starting point and the lowest point of the swing.
First, let's calculate the vertical component of the rope's length. We can use trigonometry to find this value:
Vertical component = length of rope * sin(angle)
Vertical component = 4.5m * sin(40°)
Vertical component ≈ 2.89m
Next, let's find the difference in height. Since the rope is initially at an angle of 40° relative to the vertical, the height difference will be the same as the vertical component:
Difference in height = Vertical component
Difference in height = 2.89m
Now we can calculate the change in potential energy using the formula:
Change in potential energy = mass * gravitational acceleration * difference in height
The gravitational acceleration can be approximated as 9.8 m/s².
Change in potential energy = 52kg * 9.8 m/s² * 2.89m
Change in potential energy ≈ 1513.52 Joules
Therefore, the teen's change in potential energy as he swings down and out to the point where the rope is in the vertical position is approximately 1513.52 Joules.
To calculate the change in potential energy, we need to determine the difference in the height of the teenager between the initial and final positions.
First, we need to find the vertical component of the rope's length. We can calculate this using trigonometry:
Vertical Component = length of rope * sin(angle)
= 4.5 m * sin(40°)
Now, we need to find the difference in height between the initial and final positions. As the rope is in the vertical position, the height is the same as the vertical component we just calculated.
Change in height = Vertical Component
= 4.5 m * sin(40°)
Finally, we can calculate the change in potential energy using the formula:
Change in Potential Energy = mass * gravity * change in height
Where:
mass = 52 kg (mass of the teenager)
gravity = 9.8 m/s^2 (acceleration due to gravity)
Change in Potential Energy = 52 kg * 9.8 m/s^2 * (4.5 m * sin(40°))
Now, we can calculate the numerical value of the change in potential energy.