A light spring with force constant 3.45 N/m is compressed by 7.92 cm as it is held between a 0.241 kg block on the left and a 0.482 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 ì = 0.136 . (Let the coordinate system be positive to the right and negative to the left. Be sure to include the sign to indicate the direction of the acceleration.)
ì = 0.136
heavier block = ? (m/s^2)
lighter block = ? (m/s^2)
To find the acceleration with which each block starts to move, we need to consider the forces acting on each block separately.
For the heavier block (0.482 kg), the forces acting on it are:
1. The force exerted by the spring, which pushes the block to the right.
2. The force of kinetic friction, opposing the motion and directed to the left.
Let's calculate the net force on the heavier block:
1. Force exerted by the spring = k * x (Hooke's Law, where k is the force constant and x is the displacement)
= 3.45 N/m * 0.0792 m
= 0.27384 N (directed to the right)
2. Force of kinetic friction = coefficient of kinetic friction * normal force
= 0.136 * (0.482 kg * 9.8 m/s^2)
= 0.63656 N (directed to the left)
The net force on the heavier block is the difference between these forces:
Net force = Force exerted by the spring - Force of kinetic friction
= 0.27384 N - 0.63656 N
= -0.36272 N
Since the net force is negative, the heavier block experiences a net force to the left. Using Newton's second law (F = ma), the acceleration of the heavier block can be calculated:
-0.36272 N = (0.482 kg) * a_heavier_block
a_heavier_block = -0.7524 m/s^2 (negative sign indicates acceleration to the left)
For the lighter block (0.241 kg), the forces acting on it are:
1. The force exerted by the spring, which pushes the block to the left.
2. The force of kinetic friction, opposing the motion and directed to the right.
Let's calculate the net force on the lighter block:
1. Force exerted by the spring = k * x (Hooke's Law)
= 3.45 N/m * 0.0792 m
= 0.27384 N (directed to the left)
2. Force of kinetic friction = coefficient of kinetic friction * normal force
= 0.136 * (0.241 kg * 9.8 m/s^2)
= 0.3188352 N (directed to the right)
The net force on the lighter block is the difference between these forces:
Net force = Force exerted by the spring - Force of kinetic friction
= 0.27384 N - 0.3188352 N
= -0.0449952 N
Since the net force is negative, the lighter block experiences a net force to the left. Using Newton's second law (F = ma), the acceleration of the lighter block can be calculated:
-0.0449952 N = (0.241 kg) * a_lighter_block
a_lighter_block = -0.186826 m/s^2 (negative sign indicates acceleration to the left)
Therefore, the acceleration with which each block starts to move is:
For the heavier block: a_heavier_block = -0.7524 m/s^2 (to the left)
For the lighter block: a_lighter_block = -0.186826 m/s^2 (to the left)
To find the acceleration with which each block starts to move, we need to consider the forces acting on each block separately.
For the heavier block (0.482 kg):
1. Calculate the force exerted by the spring:
The force exerted by the compressed spring is given by Hooke's Law:
F = -k * x
where F is the force, k is the force constant, and x is the displacement from the equilibrium position.
Using the given information, the force exerted by the spring is:
F = -3.45 N/m * 0.0792 m (converting cm to m) = -0.27324 N
The negative sign indicates that the force is in the opposite direction to the displacement.
2. Calculate the force of kinetic friction:
The force of kinetic friction can be calculated using the equation:
f_k = ì * N
where f_k is the force of kinetic friction, ì is the coefficient of kinetic friction, and N is the normal force.
In this case, the normal force N is equal to the weight of the block:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity.
So, N = 0.482 kg * 9.8 m/s^2 = 4.7336 N
Now, we can calculate the force of kinetic friction:
f_k = 0.136 * 4.7336 N = 0.6446976 N
3. Calculate the net force:
The net force acting on the heavier block is equal to the sum of the force exerted by the spring and the force of kinetic friction:
Net Force = F - f_k = -0.27324 N - 0.6446976 N = -0.9179376 N
Again, the negative sign indicates that the force is in the opposite direction.
4. Calculate the acceleration:
The net force is related to the mass and acceleration by Newton's second law:
Net Force = m * a
Rearranging the equation, we can solve for the acceleration:
a = Net Force / m = -0.9179376 N / 0.482 kg = -1.90219 m/s^2
So, the acceleration of the heavier block is -1.90219 m/s^2 (negative indicating it is in the leftward direction).
Now, let's calculate the acceleration of the lighter block (0.241 kg):
1. Calculate the force exerted by the spring:
Using the same equation as before, the force exerted by the compressed spring is:
F = -k * x = -3.45 N/m * 0.0792 m = -0.27324 N
2. Calculate the force of kinetic friction:
The normal force can be calculated in the same way as before:
N = m * g = 0.241 kg * 9.8 m/s^2 = 2.3638 N
Now, we can calculate the force of kinetic friction:
f_k = ì * N = 0.136 * 2.3638 N = 0.3212888 N
3. Calculate the net force:
The net force acting on the lighter block is equal to the sum of the force exerted by the spring and the force of kinetic friction:
Net Force = F - f_k = -0.27324 N - 0.3212888 N = -0.59453 N
4. Calculate the acceleration:
Using Newton's second law, we can solve for the acceleration:
a = Net Force / m = -0.59453 N / 0.241 kg = -2.4663 m/s^2
So, the acceleration of the lighter block is -2.4663 m/s^2 (negative indicating it is in the leftward direction).
In summary, the acceleration of the heavier block is -1.90219 m/s^2 (in the leftward direction) and the acceleration of the lighter block is -2.4663 m/s^2 (also in the leftward direction).