A 0.455 kg wood block is firmly attached to a very light horizontal spring (k = 160 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?
To solve this problem, we need to consider the forces acting on the block in different stages of its motion.
1. When the block is compressed:
The force exerted by the spring is given by Hooke's Law: F = -kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, x = 5.0 cm = 0.05 m and k = 160 N/m.
Therefore, the force exerted by the spring is F = -(160 N/m)(-0.05 m) = 8 N.
2. When the block is released:
As the block moves forward, the spring force acts in the opposite direction to the block's motion until the block reaches the equilibrium position. At this point, the spring force becomes zero.
The only force acting on the block is the kinetic friction force, which can be determined using Newton's second law: F_friction = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force.
Since there is no vertical acceleration, the normal force equals the weight of the block: N = m * g, where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2).
In this case, m = 0.455 kg, so the normal force N = (0.455 kg)(9.8 m/s^2) = 4.459 N.
Therefore, the friction force is F_friction = μ_k * N.
3. When the block overshoots the equilibrium position:
At this point, the spring force acts in the same direction as the block's motion.
Now, let's solve for the coefficient of kinetic friction:
1. Determine the friction force:
Since the block comes to rest and turns back after overshooting the equilibrium position, the friction force is equal in magnitude but opposite in direction to the force exerted by the spring.
F_friction = 8 N.
2. Calculate the coefficient of kinetic friction:
F_friction = μ_k * N.
8 N = μ_k * 4.459 N.
Simplifying, we find: μ_k = 8 N / 4.459 N ≈ 1.794.
Therefore, the coefficient of kinetic friction between the block and the table is approximately 1.794.
To find the coefficient of kinetic friction between the block and the table, we need to analyze the forces acting on the system.
Let's consider the block at two different positions: the compressed position (before releasing) and the maximum displacement position.
1. Compressed position:
At this position, the block is compressed by 5.0 cm. The spring force can be calculated using Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. The formula for the spring force is:
Fs = -kx
Where Fs is the spring force, k is the spring constant, and x is the displacement from the equilibrium position. Since the spring is compressed, x is negative. In this case, x = -0.05 m (converting from cm to m), and k = 160 N/m.
Fs = -(160 N/m) * (-0.05 m)
Fs = 8 N
The spring force acts in the opposite direction of the compression, so it is in the positive direction.
2. Maximum displacement position:
At this position, the block has stretched 2.3 cm beyond the equilibrium position. Again, we can use Hooke's Law to calculate the spring force, but the displacement is now positive:
Fs = -kx
Fs = -(160 N/m) * (0.023 m)
Fs = -3.68 N
The spring force acts in the opposite direction of the displacement, so it is negative.
Now let's consider the forces acting on the block when it is in motion at the maximum displacement position:
1. Weight (mg): The weight of the block is given by the formula weight = mass * gravity, where m is the mass of the block and g is the acceleration due to gravity. The mass of the block is 0.455 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight (mg) is:
mg = (0.455 kg) * (9.8 m/s^2)
mg = 4.459 N
2. Normal force (N): The normal force is the force exerted by the table, perpendicular to its surface. It is equal in magnitude and opposite in direction to the weight of the block, so N = 4.459 N.
3. Frictional force (Ff): The frictional force is given by the formula Ff = coefficient of friction * Normal force. We need to find the coefficient of kinetic friction, so let's denote it as µk. Therefore, the frictional force is:
Ff = µk * N
Since the block is moving, the frictional force acts in the direction opposite to the motion and the spring force. Hence, it is positive in this case.
Now, using the derived values, let's calculate the coefficient of kinetic friction:
Ff = 4.459 N = µk * 4.459 N
µk = 1.0
Therefore, the coefficient of kinetic friction between the block and the table is 1.0.