As part of a challenge lab, a student determines that a spring powered marble cannon will fire a 0.03kg marble to a maximum height of 0.8m when the cannon is set to the second medium firing position and fired vertically
A. If setting the medium firing position requires that the spring in the cannon is compressed by 2.5 cm, what is the spring constant of the spring?
b. If the cannon is set to it's high firing position, the spring is compressed 4cm. What is the maximum height the marble will reach when fired vertically
To solve this problem, we need to make use of the conservation of mechanical energy principle, which states that the total mechanical energy of a system, consisting of potential energy and kinetic energy, remains constant as long as no external forces (like air resistance) are acting on it.
We'll start by solving part A:
A. To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
The formula for Hooke's Law is: F = -kx
Where:
F is the force applied by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
In this case, the displacement is given as 2.5 cm, which we'll convert to meters: 2.5 cm = 0.025 m.
The force exerted by the spring in this position can be calculated using the weight and acceleration due to gravity: F = mg, where m is the mass of the marble and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the marble's mass is 0.03 kg, we can calculate the force:
F = (0.03 kg) * (9.8 m/s^2) = 0.294 N.
We can now substitute the values into Hooke's Law:
0.294 N = -k * 0.025 m.
Simplifying the equation, we find the spring constant (k):
k = -0.294 N / 0.025 m.
Therefore, the spring constant of the spring is approximately k = -11.76 N/m (negative sign indicates the direction of the force).
Now, let's move on to part B:
B. We need to calculate the maximum height the marble will reach when the cannon is set to its high firing position, where the spring is compressed by 4 cm (0.04 m).
First, we need to find the potential energy of the marble at its maximum height. At the highest point, all of its initial kinetic energy will be converted into potential energy.
The potential energy of an object is given by the equation: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Given that the mass of the marble is 0.03 kg, we can calculate the potential energy:
PE = (0.03 kg) * (9.8 m/s^2) * (0.8 m).
Therefore, the potential energy at the maximum height is PE = 0.2352 J.
Since the total mechanical energy remains constant, the potential energy at the maximum height must be equal to the initial potential energy of the spring.
The formula for potential energy stored in a spring is: PE = (1/2) * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
Rearranging the equation, we can solve for x:
2 * PE = k * x^2.
Substituting the known values:
2 * 0.2352 J = k * (0.04 m)^2.
Solving for k:
k = (2 * 0.2352 J) / (0.04 m)^2.
Therefore, the spring constant when the spring is compressed by 4 cm is approximately k = 0.294 N/m.
Note: The result is the same as in part A because in both cases, the total mechanical energy remains constant.