A(n) 160 g block is pushed by an external force against a spring with spring constant of 1260 N/m until the spring is compressed by 11.8 cm from its uncompressed length. The compressed spring and block rests at the bottom of an incline of angle θ = 39.0°. Note that the spring lies along the surface of the ramp (see Figure). Assume that the ramp is frictionless. Now, the external force is rapidly removed so that the compressed spring can push up the mass. How far up the ramp (i.e., the length along the ramp) will the block move (measured from the uncompressed end of the spring) before reversing direction and sliding back? Remember, the block is not attached to the spring. Answer in units of m. Use 9.80 m/s2 for g.
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To solve this problem, we need to consider the conservation of mechanical energy. Initially, the block is at the bottom of the incline, with the spring compressed against a force. When the external force is removed, the spring begins to push the block up the incline. The block will continue moving until it reaches a point where its potential energy is maximum and kinetic energy is minimum. At this point, the block will reverse direction and slide back down the incline.
Let's break down the solution into steps:
Step 1: Find the potential energy of the compressed spring
The potential energy stored in the compressed spring can be calculated using the formula:
Potential Energy (U) = (1/2) * k * x^2
where
k = spring constant = 1260 N/m (given)
x = compression of the spring = 11.8 cm = 0.118 m (given)
Therefore, the potential energy of the compressed spring is:
U = (1/2) * 1260 N/m * (0.118 m)^2
Step 2: Find the gravitational potential energy at the maximum height
At the point where the block comes to rest and reverses direction, it will have maximum potential energy and minimum kinetic energy. Gravitational potential energy can be calculated using the formula:
Gravitational Potential Energy (PE) = m * g * h
where
m = mass = 160 g = 0.16 kg (given)
g = acceleration due to gravity = 9.80 m/s^2 (given)
h = height from the uncompressed end of the spring to the highest point on the ramp
Step 3: Equate the potential energy of the spring and the gravitational potential energy
At the maximum height, the potential energy of the compressed spring will be equal to the gravitational potential energy. So we can equate the two equations:
(1/2) * k * x^2 = m * g * h
Step 4: Solve for h to find the distance up the ramp
Rearrange the equation from Step 3 to solve for h:
h = (1/2) * (k/m) * x^2
Substitute the known values:
h = (1/2) * (1260 N/m) / (0.16 kg) * (0.118 m)^2
Simplify and calculate the value of h.
Once you have the value of h, you can get the length along the ramp by multiplying it by the sine of the angle θ. So the final answer would be:
Length along ramp = h * sin(θ)
Plug in the values of h and θ to calculate the final answer.