A little girl of mass 30kg is jumping straight up and down on a stretched trampoline. During a single jump, she reaches a maximum height of 5.0m above the horizontal surface of the trampoline and deeps to 1.0m below the surface. Use the acceleration due to g=10m/s^2 and treat the girl as a point particle.
(a) Calculate the work done on the girl by the elastic force of the trampoline during a single fall.
(b) Now treat the trampoline as a simple spring that obeys the Hooke's law and obtain it's spring constant.
During the fall she loses 6 meters of altitude.
That means she looses m g h = 30 * 10 * 6 = 1800 Joules of potential energy
That energy went into stretching the spring
so
1800 Joules of work done
energy in spring = (1/2) k x^2 where x is spring compression or tension
so
(1/2) k (1)^2 = 1800 solve for k
To calculate the work done on the girl by the elastic force of the trampoline during a single fall, we can use the work-energy principle. The work done by a force can be calculated as the force multiplied by the displacement in the direction of the force.
(a) Work Done on the Girl during a Single Fall:
1. First, let's calculate the potential energy at the maximum height. The potential energy is given by the formula: PE = mgh, where m is the mass (30kg), g is the acceleration due to gravity (10m/s^2), and h is the height (5.0m).
PE = (30kg) × (10m/s^2) × (5.0m) = 1500 Joules
2. Next, let's calculate the potential energy at the lowest point. The potential energy at the lowest point is given by the formula: PE = mgh, where h is the depth (1.0m) below the surface.
PE = (30kg) × (10m/s^2) × (1.0m) = 300 Joules
3. The work done on the girl by the elastic force of the trampoline during a single fall is the change in potential energy, which is given by:
Work = PE at maximum height - PE at lowest point
Work = 1500 Joules - 300 Joules
Work = 1200 Joules
Therefore, the work done on the girl by the elastic force of the trampoline during a single fall is 1200 Joules.
(b) Spring Constant of the Trampoline:
To treat the trampoline as a simple spring, we can use Hooke's law, which states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position.
According to Hooke's law, the force exerted by the trampoline (F) can be written as: F = kx, where k is the spring constant and x is the displacement from the equilibrium position.
1. The displacement from the equilibrium position is the difference between the maximum height and the lowest point (5.0m - (-1.0m) = 6.0m).
2. The force required to stretch or compress the trampoline can be calculated using Newton's second law: F = ma, where m is the mass of the girl and a is the acceleration due to gravity. Since the girl is in freefall, a = g.
F = mg
F = (30kg) × (10m/s^2)
F = 300 N
3. Using Hooke's law (F = kx), we can now solve for the spring constant (k):
kx = F
k × 6.0m = 300 N
k = 300 N / 6.0m
k = 50 N/m
Therefore, the spring constant of the trampoline is 50 N/m.
To calculate the work done on the girl by the elastic force of the trampoline during a single fall, we need to first determine the total change in potential energy.
The potential energy of an object at a given height is given by the formula:
Potential energy = mass * acceleration due to gravity * height
The girl's maximum height above the horizontal surface of the trampoline is 5.0m, and she dips 1.0m below the surface. So, the total change in height is 5.0m + 1.0m = 6.0m.
Using the formula, the potential energy change is:
Potential energy = 30kg * 10m/s^2 * 6.0m = 1800 Joules
The work done on the girl by the elastic force of the trampoline during a single fall is equal to the change in potential energy. Therefore, the work done is 1800 Joules.
Now, let's move on to treating the trampoline as a simple spring that obeys Hooke's law to determine its spring constant.
Hooke's law states that the force required to stretch or compress a spring by a certain distance is directly proportional to that distance. Mathematically, it can be written as:
Force = spring constant * displacement
In this case, the displacement is the change in height of the girl, which is 6.0m.
We already know the work done by the elastic force of the trampoline, which is 1800 Joules. The work done by a spring force is given by the formula:
Work = (1/2) * spring constant * displacement^2
Plugging in the known values, we have:
1800 Joules = (1/2) * spring constant * (6.0m)^2
Simplifying the equation, we get:
spring constant = (2 * 1800 Joules) / (36.0m^2)
spring constant ≈ 100 N/m
Therefore, the spring constant of the trampoline is approximately 100 N/m.