A car is hauling a 89 kg trailer, to which it is connected by a spring. The spring constant is 2100 N/m. The car accelerates with an acceleration of 0.29 m/s2. By how much does the spring stretch?
Well, if the car is hauling a trailer, it sounds like a spring-loaded road trip! Let's crunch some numbers and stretch some springs!
To figure out how much the spring stretches, we can use Hooke's Law, which states that the force exerted by a spring is proportional to its displacement.
The formula for Hooke's Law is F = k * x, where F is the force, k is the spring constant, and x is the displacement.
In this case, the force exerted by the spring is the same as the force experienced by the trailer. We can calculate the force using Newton's second law, F = m * a, where F is the force, m is the mass, and a is the acceleration.
Now, let's plug in the values. The mass of the trailer is 89 kg, and the acceleration of the car is 0.29 m/s^2.
So, using Newton's second law: F = m * a
F = 89 kg * 0.29 m/s^2
The force exerted by the spring is equal to the force experienced by the trailer, so we can substitute F in Hooke's Law:
k * x = 89 kg * 0.29 m/s^2
Now we can solve for x, the displacement of the spring. So, dividing both sides of the equation by the spring constant (k):
x = (89 kg * 0.29 m/s^2) / 2100 N/m
Let's do some calculations:
x ≈ 0.01273809524 meters
So, it seems the spring stretches by approximately 0.0127 meters. That's not much of a stretch, but hey, every little stretch counts, right? Keep on truckin'!
To find out how much the spring stretches, we need to use Hooke's law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position.
The formula for Hooke's law is: F = k * x
Where:
F is the force applied by the spring (in Newtons),
k is the spring constant (in N/m), and
x is the displacement or stretch of the spring (in meters).
In this case, the force applied by the spring is equal to the weight of the trailer.
The weight of the trailer can be calculated using the formula: F = m * g
Where:
m is the mass of the trailer (in kilograms),
g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given:
m = 89 kg
k = 2100 N/m
g = 9.8 m/s^2
First, let's calculate the force applied by the spring (F):
F = m * g
F = 89 kg * 9.8 m/s^2
F = 872.2 N
Now, let's rearrange Hooke's law to solve for the displacement of the spring (x):
F = k * x
x = F / k
x = 872.2 N / 2100 N/m
x ≈ 0.41524 m (rounded to 5 decimal places)
Therefore, the spring stretches by approximately 0.41524 meters.
To determine how much the spring stretches, we need to calculate the force exerted by the spring. The force exerted by a spring is given by Hooke's Law, which states that the force is equal to the spring constant multiplied by the displacement.
The formula for Hooke's Law is:
F = k * x
Where:
F is the force exerted by the spring (in Newtons)
k is the spring constant (in N/m)
x is the displacement or stretch of the spring (in meters)
In this case, the force exerted by the spring is equal to the weight of the trailer. The weight of an object is given by the formula:
W = m * g
Where:
W is the weight of the object (in Newtons)
m is the mass of the object (in kilograms)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Given:
m = 89 kg
k = 2100 N/m
g = 9.8 m/s^2
First, we need to calculate the weight of the trailer:
W = m * g
W = 89 kg * 9.8 m/s^2
W = 872.2 N
Now we can calculate the displacement or stretch of the spring:
F = k * x
872.2 N = 2100 N/m * x
To solve for x, divide both sides of the equation by 2100 N/m:
x = 872.2 N / 2100 N/m
x = 0.415 m
Therefore, the spring stretches by approximately 0.415 meters when the car accelerates.
F = m*a = 89 * 0.29 = 25.8 N.
d = 25.8N/2100N * 1m = 0.0123 m.