A 3.0 kg brick rests on a perfectly smooth ramp inclined at 34 above the horizontal. The brick is kept from sliding down the plane by an ideal spring that is aligned with the surface and attached to a wall above the brick. The spring has a spring constant (force constant) of 120 N/m. By how much does the spring stretch with the brick attached?

Fb = m*g = 3kg * 9.8N/kg = 29.4 N. = Force of the brick.

Fp = 29.4*sin34 = 16.44 N. = Force parallel to the incline.

d = (16.44/120) * 1m = 0.137 m.

Well, let's see. If you want to know how much the spring stretches, you're basically asking how much the brick is pulling on the spring. And to figure that out, we need to know the force acting on the brick.

In this case, the only force acting on the brick is its weight, which is the product of its mass and the acceleration due to gravity. So, the weight of the brick can be calculated as 3.0 kg times 9.8 m/s^2.

But wait, there's more! The weight of the brick is not acting straight down, it's acting along the inclined plane. So we need to find the component of the weight that is acting parallel to the plane. That can be calculated as the weight times the sine of the angle of the plane.

Now, this force is balanced by the force exerted by the spring. According to Hooke's Law, the force exerted by a spring is equal to its spring constant times the distance it stretches. So we can set up an equation:

Force of the spring = force of the weight

120 N/m times the distance the spring stretches = (3.0 kg times 9.8 m/s^2) times the sine of 34 degrees

Now we just have to solve for the distance the spring stretches. But since I'm a clown and not a mathematician, I'll let you do the honors. Good luck!

To find how much the spring stretches with the brick attached, we can start by analyzing the forces acting on the brick on the inclined plane.

First, we need to determine the component of the weight of the brick that acts parallel to the plane. This can be found using the formula:

F_parallel = m * g * sin(Θ)

Where:
m = mass of the brick = 3.0 kg
g = acceleration due to gravity = 9.8 m/s^2
Θ = angle of the ramp = 34°

Substituting these values into the formula, we get:

F_parallel = 3.0 kg * 9.8 m/s^2 * sin(34°)
F_parallel ≈ 49.11 N

Since the brick is not sliding down the plane, the force exerted by the spring is equal in magnitude but opposite in direction to the parallel component of the weight. Thus, the force exerted by the spring is 49.11 N.

Next, we can use Hooke's Law, which states that the force exerted by the spring is proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be written as:

F_spring = k * x

Where:
F_spring = force exerted by the spring = 49.11 N
k = spring constant = 120 N/m (given)
x = displacement of the spring

Rearranging the equation, we can solve for x:

x = F_spring / k
x = 49.11 N / 120 N/m
x ≈ 0.409 m

Therefore, the spring stretches by approximately 0.409 meters when the brick is attached.

To find out how much the spring stretches with the brick attached, we need to analyze the forces acting on the brick and apply Newton's second law.

Let's start by identifying the forces involved:

1. The gravitational force acting on the brick can be calculated using the equation F_gravity = m * g, where m is the mass of the brick and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, the mass is given as 3.0 kg, so F_gravity = 3.0 kg * 9.8 m/s^2.

2. The force opposing the brick from sliding down the ramp is provided by the tension in the spring. This force can be determined using Hooke's Law, which states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed from its equilibrium position. The equation for Hooke's Law is F_spring = k * x, where F_spring is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position. In this case, the spring constant is given as 120 N/m.

3. The component of the weight of the brick parallel to the ramp is responsible for the force that wants to slide the brick down the ramp. This force can be calculated using the equation F_parallel = m * g * sin(theta), where theta is the angle of inclination of the ramp, given as 34 degrees. Make sure to convert the angle to radians by multiplying it with (pi / 180) to use it in the equation.

Now, let's set up the equation to find the displacement of the spring:

F_spring = F_parallel

k * x = m * g * sin(theta)

Plugging in the given values:

120 N/m * x = 3.0 kg * 9.8 m/s^2 * sin(34 * pi / 180)

Now we can calculate the displacement x by rearranging the equation:

x = (3.0 kg * 9.8 m/s^2 * sin(34 * pi / 180)) / 120 N/m

x = (3.0 kg * 9.8 m/s^2 * 0.559) / 120 N/m

Simplifying:

x = 1.081 m

Therefore, the spring stretches by approximately 1.081 meters when the brick is attached.