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 (μk) between the surface and the block, we need to first determine the net force acting on the block and then use it to calculate the frictional force.
1. Calculate the spring constant force:
When the spring is compressed by 4.65 cm, it exerts a force on the block given by Hooke's Law:
Fs = k * x
Fs = (113 N/m) * (0.0465 m)
Fs = 5.2485 N
2. Calculate the gravitational force:
The gravitational force acting on the block is given by:
Fg = m * g
Fg = (2.48 kg) * (9.8 m/s^2)
Fg = 24.304 N
3. Calculate the net force:
The net force acting on the block is equal to the difference between the spring force and the gravitational force:
F_net = Fs - Fg
F_net = 5.2485 N - 24.304 N
F_net = -19.0555 N
Note that the negative sign indicates that the net force opposes the motion.
4. Calculate the frictional force:
The frictional force can be calculated using the equation:
F_friction = μk * F_norm
where F_norm is the normal force acting on the block.
Since the block is on a horizontal surface and is not moving vertically, the normal force is equal to the gravitational force:
F_norm = Fg = 24.304 N
Therefore, we can rewrite the equation as:
F_friction = μk * Fg
Since the block is already in motion, the coefficient of kinetic friction can be calculated as:
μk = F_friction / Fg
Plugging in the values:
μk = (-19.0555 N) / (24.304 N)
μk ≈ -0.784
The coefficient of kinetic friction between the surface and the block is approximately -0.784.
To calculate the coefficient of kinetic friction between the surface and the block, we can use the principles of Newton's laws of motion and energy conservation.
First, let's find the force exerted by the spring on the block when it is compressed. According to Hooke's Law, the force exerted by a spring is given by F = k * x, where F is the force, k is the spring constant, and x is the displacement. Plugging in the values, we have:
F = (113 N/m) * (4.65 cm) * (1 m/100 cm) [converting cm to m]
= 0.0525 N
When the block is released, this force is responsible for its initial acceleration. Using Newton's second law, we have:
F = m * a
Solving for acceleration:
a = F / m
= 0.0525 N / 2.48 kg
= 0.0212 m/s^2
Now, let's analyze the block's motion on the rough surface. The frictional force acting on the block opposes its motion and it can be calculated using:
f = μ * N
Where f is the frictional force, μ is the coefficient of kinetic friction, and N is the normal force. The normal force is equal to the weight of the block, which is given by:
N = m * g
Plugging in the values, we have:
N = 2.48 kg * 9.8 m/s^2
= 24.304 N
Now, solving for the frictional force:
f = μ * N
= μ * 24.304 N
The frictional force can also be expressed as:
f = m * a
= 2.48 kg * 0.0212 m/s^2
= 0.0524 N
Since the block is brought to a stop by the frictional force, we can set the frictional force equal to the product of the normal force and the coefficient of kinetic friction:
μ * 24.304 N = 0.0524 N
Solving for μ:
μ = 0.0524 N / 24.304 N
= 0.0022
Therefore, the coefficient of kinetic friction between the surface and the block is approximately 0.0022.