You push a 2.0 kg block against a horizontal spring, compressing the spring by 15 cm. Then you release the block, and the spring sends it sliding across a tabletop. It stops 75 cm from where you released it. The spring constant is 200 N/m. What is the block-table coefficient of kinetic friction?
Let K = Kinetic Energy
U = Potential Energy
Wf = Work done by friction
Ug = Gravitational Potential energy
Ue = Elastic Potential Energy
Law of conservation of energy states that:
K1 + U1 + Wf= K2 + U2
Given:
mass = 2 kg
x = 0.15 m
k= 200
d = 0.75 m
Solution:
K1 = 0 (because it is not moving)
U1 = Ug + Ue (Ug =0) (Ue = (1/2)(k)(x^2) )
K2 = 0
U2 = 0
0 + 0 + (1/2)(200)(-0.15)^2 + Wf = 0 + 0
Wf = -2.25 J
Observe this. Since the block is moving to the right, the force of the friction is acting to the left.
Wf = (f)(d)cos(theta)
-2.25 = friction (0.75)(cos180)
Force of the friction = 3 Joules
To find the coefficient of friction, find the normal force acting on the block
Summation of forces along Y-axis = 0
Normal force - Weight = 0
Normal force = (2 kg)(9.8 m/s^2)= 19.6 N
Friction = (coefficent) (Normal force)
3 Joules = (coefficient) (19.6 N)
Coefficient of Kinetic Friction = 0.15306
:D
To determine the block-table coefficient of kinetic friction, we'll need to analyze the motion of the block after it is released from the compressed spring.
Let's start by calculating the spring potential energy stored in the compressed spring using the formula:
Potential Energy (PE) = (1/2) * k * x^2,
where k is the spring constant and x is the displacement of the spring (15 cm = 0.15 m in this case).
PE = (1/2) * 200 N/m * (0.15 m)^2,
PE = 4.5 J.
Next, we'll determine the initial kinetic energy of the block when it's released from the spring. Since energy is conserved, the initial kinetic energy should be equal to the potential energy of the spring.
Kinetic Energy (KE) = 4.5 J.
When the block comes to a stop, all its initial kinetic energy will be converted into work done against friction. The work done against friction can be calculated using the formula:
Work (W) = force of friction * distance.
In this case, the force of friction will be equal to the weight of the block multiplied by the coefficient of kinetic friction (μ):
Force of friction = μ * Normal force,
Weight = mass * acceleration due to gravity,
Normal force = mass * acceleration due to gravity,
where the acceleration due to gravity is 9.8 m/s^2.
Let's substitute this information into the formulas:
W = μ * (m * g) * d,
W = μ * (m * g) * d,
where:
- W is the work done against friction,
- μ is the block-table coefficient of kinetic friction,
- m is the mass of the block (2.0 kg),
- g is the acceleration due to gravity (9.8 m/s^2),
- d is the distance the block traveled after being released (75 cm = 0.75 m).
We can now solve for μ:
4.5 J = μ * (2.0 kg * 9.8 m/s^2) * 0.75 m,
4.5 J = μ * 19.6 N * 0.75 m,
4.5 J = μ * 14.7 N*m,
μ = 4.5 J / (14.7 N*m),
μ ≈ 0.306.
Therefore, the block-table coefficient of kinetic friction is approximately 0.306.
To find the block-table coefficient of kinetic friction, we first need to calculate the spring force acting on the block, which is equal to the force required to compress the spring.
The spring force can be calculated using Hooke's Law formula:
F_spring = k * Δx,
where F_spring is the spring force, k is the spring constant, and Δx is the compression or elongation of the spring.
Given:
k = 200 N/m (spring constant),
Δx = 15 cm = 0.15 m (compression of the spring).
Substituting the values into the formula, we can find the spring force:
F_spring = 200 N/m * 0.15 m = 30 N.
The spring force acts in the opposite direction to the block's motion. Thus, it provides the force necessary to overcome the kinetic friction force and bring the block to a stop.
Next, we need to calculate the work done by the spring force. The work done by a force can be found using the formula:
Work = force * distance * cos(θ),
where force is the magnitude of the force, distance is the displacement, and θ is the angle between the force and the displacement vectors.
Since the spring force and the displacement vectors are in opposite directions, the angle θ is 180 degrees, and the cos(θ) = -1.
Given:
Force = 30 N (spring force),
Distance = 75 cm = 0.75 m (displacement).
Plugging in the values into the formula, we find:
Work = 30 N * -0.75 m * -1 = 22.5 J.
The work done by the spring force is equal to the work done by friction, and it can be calculated using the equation:
Work = friction force * distance * cos(θ).
Since the block comes to a stop, the work done by friction is negative. Therefore, we can rewrite the equation as:
-22.5 J = friction force * 0.75 m * 1.
Simplifying the equation, we find:
friction force = -22.5 J / 0.75 m.
Now, we can calculate the friction force:
friction force = -30 N.
The friction force can also be expressed as the product of the coefficient of kinetic friction and the normal force:
friction force = coefficient of kinetic friction * normal force.
Since the block is on a horizontal surface, the normal force is equal to the weight of the block (mg), where m is the mass of the block and g is the acceleration due to gravity.
Given:
m = 2.0 kg (mass of the block).
The acceleration due to gravity, g, is approximately 9.8 m/s^2.
Calculating the normal force, we find:
normal force = m * g = 2.0 kg * 9.8 m/s^2 = 19.6 N.
Now, we can solve for the coefficient of kinetic friction:
coefficient of kinetic friction = friction force / normal force.
Plugging in the values, we have:
coefficient of kinetic friction = -30 N / 19.6 N = -1.53.
The negative sign indicates that the friction force acts in the opposite direction to the block's motion, which is expected.
Therefore, the block-table coefficient of kinetic friction is approximately 1.53.