A light spring with force constant 3.85 N/m is compressed by 5.00 cm as it is held between a 0.250 kg block on the left and a 0.500 kg block on the right, both resting on a horizontal surface. The spring exerts a force on each block, tending to push them apart. The blocks are simultaneously released from rest. Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is the following values. Let the coordinate system be positive to the right and negative to the left.
a)µ = 0
b)µ = 0.040
c)µ = 0.462
Find the force on the spring: kx
Then the force of friction on each block mu*mg
a=acceelrating force/m= (kx-mu*mg)/m
do that for each block.
To find the acceleration with which each block starts to move, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. We also need to consider the forces acting on each block: the force from the spring, the force of gravity, and the force of friction.
First, let's consider the case where the coefficient of kinetic friction (µ) is 0.
a) When µ = 0:
The force of gravity acting on each block can be calculated using the equation F_gravity = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For the 0.250 kg block:
F_gravity = (0.250 kg) * (9.8 m/s^2) = 2.45 N
For the 0.500 kg block:
F_gravity = (0.500 kg) * (9.8 m/s^2) = 4.90 N
The force exerted by the spring is given by Hooke's Law: F_spring = k * x, where k is the force constant of the spring and x is the compression of the spring.
F_spring = (3.85 N/m) * (0.050 m) = 0.1925 N
Since there is no friction, the only opposing force to the spring force is the force of gravity acting on each block.
For the 0.250 kg block:
net force = F_spring - F_gravity = 0.1925 N - 2.45 N = -2.2575 N
For the 0.500 kg block:
net force = F_spring - F_gravity = 0.1925 N - 4.90 N = -4.7075 N
Since the net force is negative in each case, it means that the force of the spring is not enough to overcome the force of gravity, and the blocks will not move.
b) When µ = 0.040:
In this case, there is a non-zero coefficient of kinetic friction. The force of kinetic friction (F_friction) can be calculated using the equation F_friction = µ * F_normal, where µ is the coefficient of kinetic friction and F_normal is the normal force.
The normal force (F_normal) can be calculated as the force of gravity acting on each block.
For the 0.250 kg block:
F_normal = F_gravity = 2.45 N
For the 0.500 kg block:
F_normal = F_gravity = 4.90 N
Now, we can calculate the force of kinetic friction for each block:
For the 0.250 kg block:
F_friction = (0.040) * (2.45 N) = 0.098 N
For the 0.500 kg block:
F_friction = (0.040) * (4.90 N) = 0.196 N
Again, the force exerted by the spring is 0.1925 N.
For the 0.250 kg block:
net force = F_spring - F_friction - F_gravity = 0.1925 N - 0.098 N - 2.45 N = -2.3555 N
For the 0.500 kg block:
net force = F_spring - F_friction - F_gravity = 0.1925 N - 0.196 N - 4.90 N = -4.9035 N
Since the net force is negative in each case, it means that the force of the spring is still not enough to overcome both the force of gravity and the force of friction, and the blocks will not move.
c) When µ = 0.462:
Using the same approach, we can calculate the force of kinetic friction for each block:
For the 0.250 kg block:
F_friction = (0.462) * (2.45 N) = 1.1307 N
For the 0.500 kg block:
F_friction = (0.462) * (4.90 N) = 2.2614 N
Again, the force exerted by the spring is 0.1925 N.
For the 0.250 kg block:
net force = F_spring - F_friction - F_gravity = 0.1925 N - 1.1307 N - 2.45 N = -3.3882 N
For the 0.500 kg block:
net force = F_spring - F_friction - F_gravity = 0.1925 N - 2.2614 N - 4.90 N = -7.9689 N
In this case, the net force is negative for both blocks, indicating that the force of the spring is still not enough to overcome both the force of gravity and the force of friction, and the blocks will not move.
In all the scenarios (a, b, and c), the blocks will not start to move due to the opposing forces of gravity and friction being greater than the force exerted by the spring.