Two identical massless springs of constant k = 200N/m are fixed at opposite ends of a level track, as shown. A 5 kg block is pressed against the left spring, compressing it by 0.15m. The block is then released from rest. The entire track is frictionless except for the region of length 0.25m between A and B, where mK = 0.08.
a) Determine the maximum compression of the spring on the right the first time the block arrives there.
Show the equation and then plug the numbers into that equation and then show the answer at the end
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To determine the maximum compression of the spring on the right, we can use the principle of conservation of mechanical energy.
Initially, the block is at rest and the left spring is compressed. The potential energy stored in the compressed spring is given by the equation:
PEleft = (1/2)k * x^2
where k is the spring constant and x is the compression distance.
Given that the spring constant is k = 200 N/m, and the compression distance is x = 0.15 m, the potential energy stored in the left spring is:
PEleft = (1/2) * 200 * (0.15)^2 = 4.5 J
Since the entire track is frictionless, when the block reaches the right spring, the potential energy stored in the left spring is fully converted into the kinetic energy of the block.
However, there is a frictional force acting on the block in the region between points A and B. The work done by the frictional force is given by:
Wfriction = f * d
where f is the frictional force and d is the distance between points A and B.
Given that the coefficient of kinetic friction is μk = 0.08, and the distance d = 0.25 m, we can determine the work done by friction:
Wfriction = μk * m * g * d
where m is the mass of the block and g is the acceleration due to gravity. The mass of the block is given as 5 kg, and g is approximately 9.8 m/s^2. Plugging in these values, we get:
Wfriction = 0.08 * 5 * 9.8 * 0.25 = 0.98 J
Therefore, the kinetic energy of the block when it reaches the right spring is:
KEblock = PEleft - Wfriction = 4.5 J - 0.98 J = 3.52 J
Now, the kinetic energy can be equated to the potential energy of the right spring when it is maximally compressed. The potential energy stored in a spring is given by the equation:
PEright = (1/2)k * y^2
where k is the spring constant and y is the compression distance.
Since the right spring is identical to the left spring, the spring constant is also 200 N/m. Therefore, the equation becomes:
PEright = (1/2) * 200 * y^2 = KEblock
Plugging in the value of KEblock, we can solve for y:
(y^2) = (2 * KEblock) / k
(y^2) = (2 * 3.52) / 200
y = sqrt(0.0352)
y ≈ 0.1875 m
Therefore, the maximum compression of the spring on the right when the block first reaches there is approximately 0.1875 m.