a. How much power is generated by a motor that lifts a 300.0 kg crate vertically 8.0 meters in 6.0 seconds?
b. If the crate is dropped on a large vertical spring that has a spring constant of 200,000 N/m, how far does it compress the spring?
a. To find the power generated by the motor, we need to first calculate the work done by the motor in lifting the crate. The work done, W, is given by the formula W = force × distance. In this case, the force is the weight of the crate, which is equal to its mass multiplied by the acceleration due to gravity (9.8 m/s^2).
So, force = mass × acceleration due to gravity = 300.0 kg × 9.8 m/s^2 = 2940 N.
The distance is the vertical distance the crate is lifted, which is 8.0 meters.
Now, we can calculate the work done by the motor:
W = force × distance = 2940 N × 8.0 m = 23520 J (joules)
The power, P, is defined as the rate at which work is done, or the work done per unit time. It can be calculated using the formula P = W / t, where t is the time taken.
Plugging in the values:
P = 23520 J / 6.0 s = 3920 W (watts)
Therefore, the power generated by the motor is 3920 watts.
b. To calculate how far the crate compresses the spring when dropped, we need to first calculate the potential energy stored in the spring when compressed.
The potential energy, PE, stored in a spring is given by the formula PE = 0.5 × k × x^2, where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the spring constant is 200,000 N/m. We need to find the displacement, x.
When the crate is dropped onto the spring, its potential energy is converted into the potential energy of the compressed spring.
Thus, PE of the crate = PE of the compressed spring
(mass × g × h)crate = 0.5 × k × x^2
Rearranging the equation to solve for x:
x = √((2 × (mass × g × h)crate) / k)
The mass of the crate is 300.0 kg, the acceleration due to gravity is 9.8 m/s^2, and the height is 8.0 meters.
Plugging in the values:
x = √((2 × (300.0 kg × 9.8 m/s^2 × 8.0 m)) / 200,000 N/m)
x = √((47040 Nm) / 200,000 N/m)
x = √0.2352 m^2
x = 0.485 m (rounded to three decimal places)
Therefore, the crate compresses the spring by approximately 0.485 meters.