A block of mass 2.48 kg is kept at rest as it compresses a horizontal massless spring (k= 113 N/m) by 4.65 cm. As the block is released, it travels 0.462 m on a rough horizontal surface before stopping. The acceleration of gravity is 9.8 m/s^2. Calculate the coefficient of kinetic friction between surface and block.
To find the coefficient of kinetic friction between the surface and the block, we will first calculate the force required to compress the spring and then analyze the forces acting on the block as it travels on the rough surface.
1. Calculation of the force required to compress the spring:
The force required to compress a spring is given by Hooke's law: F = k * x, where F is the force, k is the spring constant, and x is the displacement. Given the spring constant k = 113 N/m and the displacement x = 4.65 cm = 0.0465 m, we can calculate the force:
F = 113 N/m * 0.0465 m = 5.2545 N
2. Analysis of forces on the block:
When the block is released, it experiences several forces that act on it:
- The force exerted by the compressed spring, which we calculated to be 5.2545 N.
- The gravitational force pulling the block downwards, given by F_gravity = m * g, where m is the mass and g is the acceleration due to gravity. Given the mass m = 2.48 kg and g = 9.8 m/s^2, we can calculate the gravitational force:
F_gravity = 2.48 kg * 9.8 m/s^2 = 24.304 N
Since the block is at rest initially, the force exerted by the compressed spring must be equal to the force of friction for the block to start moving. Therefore, the force of friction is 5.2545 N.
3. Calculation of the coefficient of kinetic friction:
The force of friction can be calculated using the equation F_friction = μ_k * F_normal, where F_friction is the force of friction, μ_k is the coefficient of kinetic friction, and F_normal is the normal force. In this case, the normal force is equal to the gravitational force (F_gravity) since the block is on a horizontal surface.
5.2545 N = μ_k * 24.304 N
Solving for μ_k, we have:
μ_k = 5.2545 N / 24.304 N = 0.216
Therefore, the coefficient of kinetic friction between the surface and the block is approximately 0.216.
To calculate the coefficient of kinetic friction between the surface and the block, we can use Newton's laws of motion. Let's break down the problem step by step.
Step 1: Calculate the potential energy stored in the compressed spring.
The potential energy stored in a spring is given by the formula: PE = 1/2 * k * x^2, where PE is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
Given:
k = 113 N/m (spring constant)
x = 4.65 cm = 0.0465 m (displacement of the spring)
Calculating the potential energy stored in the spring:
PE = 1/2 * 113 N/m * (0.0465 m)^2
PE ≈ 0.149 J
Step 2: Calculate the work done by friction.
The work done by friction is equal to the negative of the work done by external forces. In this case, the external work done is the work done against the gravitational force.
The work done against gravity is given by the formula: W = m * g * d, where W is the work done, m is the mass, g is the acceleration due to gravity, and d is the distance traveled.
Given:
m = 2.48 kg (mass)
g = 9.8 m/s^2 (acceleration due to gravity)
d = 0.462 m (distance traveled)
Calculating the work done by gravity:
W = 2.48 kg * 9.8 m/s^2 * 0.462 m
W ≈ 11.434 J
Since there is no other external work done on the block, the work done by friction is equal to the negative of the work done against gravity.
Work done by friction = -11.434 J
Step 3: Calculate the work done by friction using the work-energy principle.
According to the work-energy principle, the work done by friction is equal to the change in kinetic energy of the block.
The change in kinetic energy of the block is given by the formula: ΔKE = 1/2 * m * (vf^2 - vi^2), where ΔKE is the change in kinetic energy, m is the mass, vf is the final velocity, and vi is the initial velocity.
Since the block started from rest, the initial velocity (vi) is 0 m/s.
Given:
m = 2.48 kg (mass)
vf = 0 m/s (final velocity)
Calculating the change in kinetic energy:
ΔKE = 1/2 * 2.48 kg * (0 - 0^2)
ΔKE = 0 J
The work done by friction is equal to the change in kinetic energy, so the work done by friction is also 0 J.
Step 4: Calculate the coefficient of kinetic friction.
The work done by friction is also given by the equation: W = f * d, where f is the force of friction and d is the distance traveled.
Given:
W = -11.434 J (work done by friction)
d = 0.462 m (distance traveled)
Calculating the force of friction:
-11.434 J = f * 0.462 m
f ≈ -24.773 N
The negative sign indicates that the force of friction acts in the opposite direction of the displacement.
The coefficient of kinetic friction (μk) can be calculated using the formula: f = μk * N, where N is the normal force.
The normal force (N) can be calculated using Newton's second law: N = m * g, where m is the mass and g is the acceleration due to gravity.
Given:
m = 2.48 kg (mass)
g = 9.8 m/s^2 (acceleration due to gravity)
Calculating the normal force:
N = 2.48 kg * 9.8 m/s^2
N ≈ 24.304 N
Calculating the coefficient of kinetic friction:
-24.773 N = μk * 24.304 N
μk ≈ -1.02
Since the coefficient of kinetic friction cannot be negative, the coefficient of kinetic friction between the surface and the block is approximately 1.02.