A 500g block is released from rest and slides down a frictionless track that begins 2m above the horizontal. At the bottom of the track, where the surface is horizontal, the block strikes and sticks to a light spring with a spring constant of 20.0N/m. Find the maximum distance the spring is compressed.
Kinetic energy of block at bottom = m g h
that goes into compressing the spring
m g h = (1/2) k x^2
solve for x^2 and x
10
To find the maximum distance the spring is compressed, we can use the principle of conservation of mechanical energy.
First, let's calculate the potential energy (PE) of the block at the starting position:
PE = m * g * h
Where:
m = mass of the block = 500g = 0.5kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the track above the horizontal surface = 2m
PE = 0.5kg * 9.8 m/s^2 * 2m
PE = 9.8 Joules
Since the track is frictionless, the PE at the starting position will be converted into the potential energy of the compressed spring (PE_spring) when the block reaches the bottom of the track. The potential energy of the compressed spring can be calculated using the equation:
PE_spring = (1/2) * k * x^2
Where:
k = spring constant = 20 N/m
x = maximum compression distance of the spring (what we want to find)
Setting the two potential energies equal to each other, we have:
PE = PE_spring
9.8 Joules = (1/2) * 20 N/m * x^2
Simplifying the equation:
4.9 = 10 * x^2
x^2 = 4.9 / 10
x^2 = 0.49
x = sqrt(0.49)
x = 0.7 m
Therefore, the maximum distance the spring is compressed is 0.7 meters.
To find the maximum distance the spring is compressed, we need to consider the conservation of mechanical energy.
The initial mechanical energy of the block is equal to its potential energy at the starting height. The final mechanical energy is equal to the potential energy at the maximum compression of the spring.
The potential energy of an object of mass m at a height h is given by the equation:
PE = mgh
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
In this case, the initial potential energy of the block is:
PE_initial = 0.5 kg * 9.8 m/s^2 * 2 m
PE_initial = 9.8 J
Since the track is frictionless and there are no other external forces acting on the block, the conservation of energy tells us that the final mechanical energy is equal to the initial mechanical energy.
The final mechanical energy is equal to the potential energy at the maximum compression of the spring:
PE_final = 0.5 * k * x^2
where k is the spring constant and x is the compression distance.
Setting the initial and final potential energy equal to each other, we have:
PE_initial = PE_final
9.8 J = 0.5 * 20 N/m * x^2
x^2 = (9.8 J) / (0.5 * 20 N/m)
Simplifying, we get:
x^2 = 0.49 m^2
Taking the square root of both sides, we find:
x = 0.7 m
Therefore, the maximum distance the spring is compressed is 0.7 meters.