A block of mass 12.0 kg slides from rest down a frictionless 35.0° incline and is stopped by a strong spring with k = 1.50 104 N/m. The block slides 3.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?
the energy going into the spring= m*g*3/sin35
that has to equal 1/2 k x^2 solve for x
To find how far the spring has been compressed when the block comes to rest, we can use the principle of conservation of mechanical energy.
First, let's consider the initial potential energy of the block at the top of the incline. The potential energy is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
The height of the incline can be found using the formula: h = d * sin(theta), where d is the distance the block slides down the incline and theta is the angle of the incline.
h = 3.00 m * sin(35.0°)
h ≈ 1.715 m
Now, we can calculate the initial potential energy:
PE_initial = m * g * h
PE_initial = 12.0 kg * 9.8 m/s^2 * 1.715 m
PE_initial ≈ 202.07 J
Next, we need to calculate the final potential energy of the block when it is stopped by the spring. At this point, all of the potential energy has been converted into elastic potential energy stored in the spring.
The formula for elastic potential energy is: PE_spring = (1/2) * k * x^2, where k is the spring constant and x is the compression of the spring.
Rearranging the formula, we can solve for x:
x = sqrt((2 * PE_spring) / k)
Substituting the values:
PE_spring = PE_initial = 202.07 J (since mechanical energy is conserved)
k = 1.50 * 10^4 N/m (given in the problem)
x = sqrt((2 * 202.07) / (1.50 * 10^4))
x ≈ 0.0277 m
Therefore, when the block comes to rest against the spring, the spring has been compressed by approximately 0.0277 m.
To find the distance the spring has been compressed, we need to calculate the potential energy stored in the spring at the point the block comes to rest.
1. Determine the gravitational potential energy:
The gravitational potential energy is given by the formula: PE_gravity = mgh, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height through which the block has fallen.
First, we need to find the vertical height h. Using trigonometry, we can calculate h as follows:
h = d * sin(θ), where d is the distance the block has slid down the incline and θ is the angle of the incline.
Plugging in the values:
d = 3.00 m
θ = 35.0°
h = 3.00 m * sin(35.0°) = 1.72 m
Now, we can calculate the gravitational potential energy:
PE_gravity = 12.0 kg * 9.8 m/s^2 * 1.72 m = 199.6 J
2. Determine the elastic potential energy:
The elastic potential energy is given by the formula: PE_elastic = 0.5 * k * x^2, where k is the spring constant and x is the distance the spring has been compressed.
Rearranging the formula:
x^2 = 2 * PE_elastic / k
Plugging in the values:
k = 1.50 * 10^4 N/m (given)
PE_elastic = PE_gravity = 199.6 J (calculated earlier)
x^2 = (2 * 199.6 J) / (1.50 * 10^4 N/m)
x^2 = 0.0266 m^2
Taking the square root of both sides:
x = sqrt(0.0266 m^2) = 0.163 m
Therefore, the spring has been compressed by approximately 0.163 meters.