A block of mass 13.9 kg slides from rest down a frictionless 40.0° incline and is stopped by a strong spring with a spring constant 32.3 kN/m (note the unit). The block slides 7.80 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest momentarily, how far has the spring been compressed? Answer in units of cm and use 9.8 m/s2 for g.

Question

A block of mass 13.9 kg slides from rest down a frictionless 40.0° incline and is stopped by a strong spring with a spring constant 32.3 kN/m (note the unit). The block slides 7.80 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest momentarily, how far has the spring been compressed? Answer in units of cm and use 9.8 m/s2 for g.

Answer 1

To find the distance the spring has been compressed, we need to consider the work done by the gravitational force and the work done by the spring force.

First, let's find the work done by the gravitational force. The work done by gravity is given by the formula:

Work_gravity = m * g * h

where m is the mass, g is the acceleration due to gravity, and h is the vertical distance the block descends along the incline.

We can find h using trigonometry. The vertical distance h is given by:

h = d * sin(θ)

where d is the distance the block slides and θ is the angle of the incline.

Now, we can find the work done by the gravitational force:

Work_gravity = m * g * h = m * g * d * sin(θ)

Next, let's find the work done by the spring force. The work done by the spring force is given by the formula:

Work_spring = (1/2) * k * x^2

where k is the spring constant and x is the displacement of the spring.

We can rearrange the equation for the work done by the spring force to solve for x:

x^2 = (2 * Work_spring) / k

Finally, we can substitute the values given in the problem and calculate the answer.

m = 13.9 kg
g = 9.8 m/s^2
θ = 40.0°
d = 7.80 m
k = 32.3 kN/m = 32,300 N/m

First, let's find h using the formula h = d * sin(θ):

h = 7.80 m * sin(40.0°)
≈ 5.01 m

Next, let's find the work done by the gravitational force:

Work_gravity = m * g * h
= 13.9 kg * 9.8 m/s^2 * 5.01 m
≈ 686 J

Now, let's find the displacement x of the spring:

x^2 = (2 * Work_spring) / k
x^2 = (2 * 686 J) / 32,300 N/m
x^2 ≈ 0.042 m^2

Finally, we convert the displacement x to centimeters by multiplying by 100:

x_cm = x * 100
≈ 4.2 cm

Therefore, when the block comes to rest momentarily, the spring has been compressed approximately 4.2 cm.