A 0.535 kg wood block is firmly attached to a very light horizontal spring (k = 175 N/m. It is noted that the block-spring system, when compressed 5.0 cm and released, stretches out 2.3 cm beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?
Figure the energy lost in the spring:
energy lost to friction: 1/2k(.05^2-.023^2)
Now the block moved 7.3cm, or .073m.
energy lost to fricion= mu*mg*.073
solve for mu.
To find the coefficient of kinetic friction between the block and the table, we can use the concept of energy conservation.
1. First, let's calculate the potential energy stored in the compressed spring.
Potential energy in the spring = (1/2) * k * x^2
where k is the spring constant and x is the compression distance.
Given: k = 175 N/m and x = 5.0 cm (converted to meters = 0.05 m)
Potential energy in the spring = (1/2) * 175 N/m * (0.05 m)^2 = 0.21875 J
2. Next, let's calculate the total mechanical energy of the block-spring system when it is released.
The total mechanical energy is the sum of the potential energy and the kinetic energy.
Total mechanical energy = Potential energy + Kinetic energy
3. At the equilibrium position, all of the potential energy is converted into kinetic energy.
So, at the equilibrium position, the total mechanical energy is equal to the kinetic energy.
Kinetic energy at the equilibrium position = Total mechanical energy = 0.21875 J
4. Now, let's calculate the maximum displacement of the block from the equilibrium position.
The block stretches to 2.3 cm (converted to meters = 0.023 m) beyond the equilibrium position.
Maximum displacement = 0.023 m
5. At the maximum displacement, all of the initial potential energy is converted into kinetic energy.
The spring potential energy at the maximum displacement is zero because it has all been converted to kinetic energy.
So, at the maximum displacement, the kinetic energy is equal to the initial potential energy.
Kinetic energy at the maximum displacement = 0.21875 J
6. Now, let's calculate the work done by kinetic friction.
The work done by kinetic friction is equal to the change in mechanical energy.
Work done by kinetic friction = Kinetic energy at the maximum displacement - Kinetic energy at the equilibrium position
Work done by kinetic friction = 0.21875 J - 0 J = 0.21875 J
7. The work done by kinetic friction can be written as the product of the force of friction and the displacement.
Work done by kinetic friction = force of friction * displacement
Since the displacement is the negative of the compression distance, the force of friction is negative.
Therefore, the work done by kinetic friction is the negative of the force of friction.
8. Now, let's calculate the force of friction.
Force of friction = - (Work done by kinetic friction / displacement)
Force of friction = - (0.21875 J / 0.023 m) = -9.5125 N
9. Finally, let's calculate the coefficient of kinetic friction.
The force of friction can be written as the product of the coefficient of kinetic friction and the normal force.
Normal force = mass * acceleration due to gravity
Normal force = 0.535 kg * 9.8 m/s^2 = 5.243 N
Therefore, the coefficient of kinetic friction = Force of friction / Normal force
Coefficient of kinetic friction = (-9.5125 N) / (5.243 N) = -1.812
Note: The negative sign indicates that the force of friction opposes the motion of the block. Since the coefficient of friction cannot be negative, we take the absolute value of the coefficient of kinetic friction, which is 1.812.
To find the coefficient of kinetic friction between the block and the table, we need to determine the acceleration of the block when it starts to move and compare it with the acceleration due to the spring.
First, let's calculate the spring constant (k) in N/m:
k = 175 N/m
Next, we can determine the force exerted by the spring when it is compressed by 5.0 cm (0.05 m):
F_spring = k * compression
= 175 N/m * 0.05 m
= 8.75 N
According to Hooke's Law, the force exerted by the spring is equal to the force required to overcome friction and accelerate the block:
F_friction = F_spring
Since we know the mass of the block (m) is 0.535 kg, we can use Newton's second law of motion to determine the acceleration due to friction:
F_friction = m * a_friction
Since the spring stretches out 2.3 cm (0.023 m) beyond the equilibrium position, we can calculate the force exerted by the spring after displacement is added:
F_spring_after_displacement = k * displacement
= 175 N/m * 0.023 m
= 4.025 N
Now, we can calculate the net force acting on the block when it is released:
Net force = F_spring_after_displacement - F_friction
= 4.025 N - 8.75 N
= -4.725 N
(Here, the negative sign indicates that the force is acting in the opposite direction of the positive x-axis)
The net force is also equal to the mass of the block multiplied by the acceleration:
Net force = m * a_total
Since the block eventually comes to a stop, the acceleration due to the spring must be equal and opposite to the acceleration due to friction. Thus, we have:
m * a_friction = m * a_spring
Using a_friction as a positive value and a_spring as a negative value, we can say:
a_friction = -a_spring
Plugging in the known values:
a_friction = -a_spring
= -4.725 N / 0.535 kg
= -8.815 m/s^2
Now, we can calculate the coefficient of kinetic friction (μ_k) using the formula:
a_friction = μ_k * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2):
-8.815 m/s^2 = μ_k * 9.8 m/s^2
Simplifying the equation:
-μ_k = -8.815 m/s^2 / 9.8 m/s^2
μ_k = 8.815 m/s^2 / 9.8 m/s^2
Therefore, the coefficient of kinetic friction between the wood block and the table is approximately 0.9.