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 = 4.00 104 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?
Let the compression of the spring be x.
When the block comes to rest, the potential energy loss equals (1/2) k x^2
The P.E. loss is M*g*3.00 sin35 = 202.4 J
Solve
202.4 = (1/2)kx^2 for x.
To determine how far the spring has been compressed when the block comes to rest, we need to find the potential energy stored in the spring.
Step 1: Find the gravitational potential energy of the block at the point where it comes to rest.
The gravitational potential energy is given by the equation:
PE = mgh
Where m is the mass of the block, g is the acceleration due to gravity, and h is the vertical height of the block.
Given:
m = 12.0 kg
g = 9.8 m/s^2 (approximate value)
h = vertical height of the block = (3.00 m)sin(35.0°) = 1.71 m (approximate value)
PE = (12.0 kg)(9.8 m/s^2)(1.71 m)
PE ≈ 199.622 J
Step 2: Find the amount of potential energy that is converted into elastic potential energy in the spring.
The potential energy stored in a spring is given by the equation:
PE = (1/2)kx^2
Where k is the spring constant and x is the displacement of the spring.
Given:
k = 4.00 × 10^4 N/m
PE = 199.622 J
199.622 J = (1/2)(4.00 × 10^4 N/m)x^2
Step 3: Solve for x, the displacement of the spring.
Rearranging the equation, we have:
x^2 = (2PE)/(k)
x^2 = (2)(199.622 J) / (4.00 × 10^4 N/m)
x^2 = 3.992 J/( 4.00 × 10^4 N/m)
Taking the square root of both sides, we have:
x ≈ √(3.992 J/( 4.00 × 10^4 N/m))
x ≈ √(0.0000998 m^2)
Therefore, the spring has been compressed by approximately 0.01 m (or 1 cm).
To find the distance the spring has been compressed, we need to calculate the potential energy stored in the spring when the block comes to rest.
Step 1: Calculate the gravitational potential energy converted to spring potential energy.
The gravitational potential energy converted to spring potential energy can be calculated using the formula:
Potential Energy = mgh
where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the incline.
Given:
Mass of the block (m) = 12.0 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Height of the incline (h) = ? (We need to find this)
We can calculate the height of the incline using trigonometry. The height of the incline is the vertical component of the distance the block slides down the incline.
Height of the incline (h) = distance * sin(angle)
Given:
Distance (d) = 3.00 m
Angle of the incline (theta) = 35.0 degrees
Using the given values, we can calculate the height of the incline:
h = 3.00 m * sin(35.0 degrees)
Step 2: Calculate the potential energy stored in the spring.
The potential energy stored in the spring can be calculated using the formula:
Potential Energy = 0.5 * k * x^2
where k is the spring constant and x is the distance the spring has been compressed.
Given:
Spring constant (k) = 4.00 * 10^4 N/m
Distance the spring has been compressed (x) = ? (We need to find this)
We can rearrange the formula to solve for x:
x = sqrt((2 * Potential Energy) / k)
The potential energy can be calculated by equating it to the gravitational potential energy converted to spring potential energy:
Potential Energy = mgh
Given:
m = 12.0 kg
g = 9.8 m/s^2
h = calculated in Step 1
Now, substitute the known values into the equation and solve for x:
Potential Energy = mgh = (12.0 kg) * (9.8 m/s^2) * h
Finally, substitute the calculated value of the potential energy and the spring constant into the formula for x:
x = sqrt((2 * Potential Energy) / k)
Calculate the value of x to find the distance that the spring has been compressed.