A 16 kg block slides down a 30 degree frictionless incline where it is stopped by a strong Hook's Law spring at the
bottom. The spring constant is k = 9.8 x10^4 N/m. If the block slides 9 m from the point where it was released (from
rest) to the point where it comes to rest against the spring, how far (in mm) has the spring been compressed?
To find the compression of the spring, we need to calculate the net work done on the block as it slides down the incline and comes to rest against the spring.
1. First, we need to find the gravitational force acting on the block. The gravitational force is given by the formula:
F_gravity = m * g,
where m is the mass of the block (16 kg) and g is the acceleration due to gravity (9.8 m/s^2).
F_gravity = 16 kg * 9.8 m/s^2
= 156.8 N.
2. Next, we need to find the component of the gravitational force that acts along the direction of motion down the incline. This can be calculated using trigonometry:
F_parallel = F_gravity * sin(θ),
where θ is the angle of the incline (30 degrees).
F_parallel = 156.8 N * sin(30 degrees)
= 78.4 N.
3. The work done by the component of gravitational force parallel to the incline is given by:
W_parallel = F_parallel * d,
where d is the distance moved by the block along the incline (9 m).
W_parallel = 78.4 N * 9 m
= 705.6 J.
4. The work done by the spring is given by the formula:
W_spring = (1/2) * k * x^2,
where k is the spring constant (9.8 x 10^4 N/m) and x is the compression of the spring.
We need to find x, so we rearrange the equation:
x = √(2 * W_spring / k).
We know the work done by the spring is equal to the work done by the component of gravitational force parallel to the incline:
W_spring = W_parallel = 705.6 J.
x = √(2 * 705.6 J / (9.8 x 10^4 N/m))
= √(1.4317 x 10^-2 m^2)
≈ 0.12 m.
5. Finally, we need to convert the distance x to millimeters:
x_mm = 0.12 m * 1000 mm/m
= 120 mm.
Therefore, the spring has been compressed by approximately 120 mm.
To find the distance the spring has been compressed, we first need to determine the potential energy of the block when it comes to rest against the spring. Since the block slides down a frictionless incline, it only loses potential energy and gains kinetic energy.
1. Calculate the initial potential energy (PE_initial) of the block at the top of the incline:
PE_initial = m * g * h
where m is the mass of the block (16 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the incline.
Since the incline is at a 30-degree angle, we can calculate the height as follows:
h = 9 m * sin(30 degrees)
2. Calculate the final potential energy (PE_final) of the block when it comes to rest against the spring. At this point, all of the block's initial kinetic energy will be converted into potential energy stored in the spring.
PE_final = 1/2 * k * x²
where k is the spring constant (9.8 x 10^4 N/m) and x is the compression of the spring.
3. Equate the initial and final potential energies to find the value of x:
PE_initial = PE_final
m * g * h = 1/2 * k * x²
4. Rearrange the equation to solve for x:
x = sqrt(2 * m * g * h / k)
5. Plug in the values:
x = sqrt(2 * 16 kg * 9.8 m/s² * 9 m * sin(30 degrees) / (9.8 x 10^4 N/m))
6. Calculate the value of x in millimeters (mm):
x_mm = x * 1000
Now you can plug in the values and calculate the distance the spring has been compressed.