An ore car of mass 37000 kg starts from rest
and rolls downhill on tracks from a mine. At
the end of the tracks, 28 m lower vertically, is
a horizontally situated spring with constant
2.8 × 10
5
N/m.
The acceleration of gravity is 9.8 m/s
2
.
Ignore friction.
How much is the spring compressed in stopping the ore car?
To find out how much the spring is compressed, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system is conserved if there are no external forces acting on it.
1. First, let's find the potential energy of the ore car at the top of the hill. The potential energy is given by the formula:
Potential Energy (PE) = mass * gravity * height
Here, the mass of the ore car is given as 37000 kg, gravity is 9.8 m/s^2, and the height is 28 m. So, the potential energy at the top of the hill is:
PE1 = 37000 kg * 9.8 m/s^2 * 28 m
2. Next, let's find the potential energy of the ore car when it reaches the spring. At this point, all of the car's potential energy has been converted into potential energy stored in the compressed spring. The potential energy of the spring is given by the formula:
Potential Energy (PE) = (1/2) * spring constant * (compression)^2
Here, the spring constant is given as 2.8 * 10^5 N/m. To find the compression, we need to use the formula for potential energy:
PE2 = (1/2) * (2.8 * 10^5 N/m) * (compression)^2
3. Since mechanical energy is conserved, we can equate the potential energy at the beginning to the potential energy at the end:
PE1 = PE2
This gives us the equation:
37000 kg * 9.8 m/s^2 * 28 m = (1/2) * (2.8 * 10^5 N/m) * (compression)^2
4. Now we can solve for the compression. Rearranging the equation, we get:
compression = √((2 * (37000 kg * 9.8 m/s^2 * 28 m)) / (2.8 * 10^5 N/m))
Plugging in the values and evaluating the expression will give us the compression of the spring.
By following these steps, you should be able to determine the amount by which the spring is compressed in order to stop the ore car.