A block of mass m = 5.2 kg is released from rest at a height of H = 4.0 m on a curved frictionless ramp. At the foot of the ramp is a spring whose spring constant is k = 192.0 N/m. What is the maximum compression of the spring, x?
After the spring reaches its maximum compression, it pushes the block backwards. What is the maximum height reached by the block?
Thank's!
PE ->KE
KE ->PE(spring) =>
PE= PE(spring)
mgh=kx²/2
x=sqrt(2mgh/k)=…
PE(spring) -> KE->PE => the same ‘h’
(due to the absence of friction)
Well, this problem sounds like a real spring-loaded roller coaster! Let's dive into it, shall we?
First, let's calculate the potential energy of the block when it's at the top of the ramp. Using the formula P.E. = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height, we get:
P.E. = (5.2 kg)(9.8 m/s²)(4.0 m) = 202.24 joules
Now, as the block slides down the ramp, this potential energy gets converted into the elastic potential energy stored in the spring. That's given by the formula U = 0.5kx², where k is the spring constant and x is the compression of the spring.
So, we have:
202.24 J = 0.5(192 N/m)(x²)
Now, we can solve for x:
404.48 J = 192 N/m(x²)
x² = 404.48 J / (192 N/m)
x² = 2.107 joules per newton
x = √(2.107 joules per newton)
So, the maximum compression of the spring, x, is approximately √(2.107 joules per newton).
Now, let's find out the maximum height reached by the block after the spring pushes it backward. The energy conservation principle tells us that the total mechanical energy (kinetic plus potential energy) is conserved.
Since the block reaches its maximum height, all of the potential energy is converted back into kinetic energy.
So, using the equation K.E. = 0.5mv², where v is the velocity, we can set the kinetic energy equal to the initial potential energy:
0.5mv² = 202.24 J
Solving for v:
v = √(404.48 J / 5.2 kg)
With the velocity known, we can now calculate the maximum height reached. Using the equation P.E. = mgh again, this time for the maximum height, we have:
202.24 J = (5.2 kg)(9.8 m/s²)(h)
h = 202.24 J / (5.2 kg)(9.8 m/s²)
So, the maximum height reached by the block is approximately 4.0 meters (ignoring any rounding errors).
I hope these calculations didn't make your head spin like a circus acrobat! If you have any more physics puzzles, feel free to ask!
To find the maximum compression of the spring, we can use the conservation of mechanical energy. The initial potential energy at the top of the ramp is equal to the sum of the final potential energy at the maximum height and the potential energy stored in the compressed spring.
The potential energy at the top of the ramp is given by m * g * H, where m is the mass of the block, g is the acceleration due to gravity, and H is the height of the ramp.
The potential energy at the maximum height is given by m * g * (H - x), where x is the compression of the spring at its maximum.
The potential energy stored in the spring is given by (1/2) * k * x^2, where k is the spring constant and x is the compression of the spring.
Setting the initial potential energy equal to the sum of the final potential energy and the potential energy stored in the spring, we have:
m * g * H = m * g * (H - x) + (1/2) * k * x^2
Simplifying the equation, we have:
m * g * H = m * g * H - m * g * x + (1/2) * k * x^2
m * g * x = (1/2) * k * x^2
2 * m * g = k * x
x = (2 * m * g) / k
Substituting the given values, we have:
x = (2 * 5.2 kg * 9.8 m/s²) / 192 N/m
x = 0.536 m
Therefore, the maximum compression of the spring is 0.536 m.
To find the maximum height reached by the block, we can use the conservation of mechanical energy again. The potential energy at the maximum height is equal to the sum of the initial potential energy at the top of the ramp and the potential energy stored in the compressed spring.
m * g * (H - x) = m * g * H + (1/2) * k * x^2
Simplifying the equation, we have:
m * g * H - m * g * x = m * g * H + (1/2) * k * x^2
- m * g * x = (1/2) * k * x^2
2 * m * g = k * x
x = (2 * m * g) / k
Substituting the given values, we have:
x = (2 * 5.2 kg * 9.8 m/s²) / 192 N/m
x = 0.536 m
Therefore, the maximum height reached by the block is also 0.536 m.
To find the maximum compression of the spring, we can start by analyzing the potential energy at the top of the ramp and equating it to the potential energy at the maximum compression of the spring.
1. Calculate the potential energy at the top of the ramp:
At the top of the ramp, only potential energy is present. The potential energy is given by the formula PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the block.
PE_top = mgh = 5.2 kg * 9.8 m/s^2 * 4.0 m = 203.84 J
2. Find the maximum compression of the spring:
At the maximum compression of the spring, all the potential energy at the top of the ramp will be converted into potential energy stored in the spring.
The potential energy stored in a spring is given by the formula PE_spring = (1/2)kx^2, where k is the spring constant and x is the compression of the spring.
Therefore, we can equate the potential energy at the top of the ramp to the potential energy stored in the spring and solve for x:
PE_top = PE_spring
mgh = (1/2)kx^2
x = sqrt(2*mgh/k)
Plugging in the given values:
x = sqrt(2 * 5.2 kg * 9.8 m/s^2 * 4.0 m / 192.0 N/m)
x ≈ sqrt(0.542)
x ≈ 0.737 m (rounded to three decimal places)
Answer: The maximum compression of the spring, x, is approximately 0.737 meters.
To find the maximum height reached by the block, we can use the principle of conservation of energy.
3. Find the maximum height reached by the block:
At the maximum compression of the spring, all the potential energy stored in the spring will be converted into potential energy of the block at its maximum height.
The potential energy of the block at its maximum height is given by the formula PE_max = mgh_max, where h_max is the maximum height reached by the block.
Since energy is conserved, we can equate the potential energy stored in the spring to the potential energy of the block at its maximum height:
PE_spring = PE_max
(1/2)kx^2 = mgh_max
h_max = (1/2)(k/m)x^2
Plugging in the given values:
h_max = (1/2)(192.0 N/m)/(5.2 kg)(0.737 m)^2
h_max ≈ 0.479 m (rounded to three decimal places)
Answer: The maximum height reached by the block, h_max, is approximately 0.479 meters.