A 209 kg block is released at height h =
3.3 m as shown. The track is frictionless
except for a portion of length 7 m. The block
travels down the track, hits a spring of force
constant k = 1545 N/m, and compresses it
1.9 m from its equilibrium position before
coming to rest momentarily.
The acceleration of gravity is 9.8 m/s
Determine the coefficient of kinetic friction
between surface and block over the 7 m track
length.
To determine the coefficient of kinetic friction between the surface and the block over the 7 m track length, we can use the conservation of energy principle.
1. Calculate the potential energy at the initial height h.
Potential energy at height h = m*g*h
Potential energy at height 3.3 m = 209 kg * 9.8 m/s^2 * 3.3 m
2. Calculate the potential energy at the compressed position of the spring.
Potential energy at the compressed position = (1/2) * k * x^2
Potential energy at 1.9 m compression = (1/2) * 1545 N/m * (1.9 m)^2
3. Calculate the loss of potential energy due to the 7 m track length.
Loss of potential energy = Potential energy at height h - Potential energy at the compressed position
4. Calculate the work done by friction.
Work done by friction = Loss of potential energy
5. Use the work done by friction to calculate the force of friction.
Force of friction = Work done by friction / distance
Force of friction = Work done by friction / 7 m
6. Use the force of friction to calculate the normal force.
Normal force = m * g
7. Calculate the coefficient of kinetic friction.
Coefficient of kinetic friction = Force of friction / Normal force
Substituting the values given:
Mass (m) = 209 kg
Gravity (g) = 9.8 m/s^2
Distance (d) = 7 m
Spring constant (k) = 1545 N/m
Compression (x) = 1.9 m
Height (h) = 3.3 m
Let's plug in the values and calculate the coefficient of kinetic friction.
To determine the coefficient of kinetic friction between the surface and the block over the 7 m track length, we need to analyze the forces acting on the block.
First, let's calculate the gravitational potential energy of the block when it is released at a height h = 3.3 m. The formula for gravitational potential energy is given by:
Potential Energy = mass * gravity * height
Potential Energy = 209 kg * 9.8 m/s^2 * 3.3 m = 6479.14 J
Next, let's consider the work done by the block due to the frictional force over the 7 m track length. The work done by friction is expressed as the product of the force of friction and the displacement:
Work = Force * Displacement
Since the block comes to rest momentarily at the end of the track, its kinetic energy is converted into potential energy stored in the spring. Therefore, the work done by friction is equal to the change in potential energy of the block:
Work = -Change in Potential Energy
Change in Potential Energy = Final Potential Energy - Initial Potential Energy
= 0 J - 6479.14 J
= -6479.14 J
Now, let's calculate the work done by friction over the 7 m track length using the work-energy theorem:
Work = -6479.14 J
We know that the work done by friction can be calculated as the product of the frictional force and the displacement:
Work = Force of Friction * Displacement
Since the displacement is 7 m, we have:
-6479.14 J = Force of Friction * 7 m
Solving for the force of friction gives us:
Force of Friction = -6479.14 J / 7 m
= -927.02 N
The negative sign indicates that the force of friction acts in the opposite direction to the motion of the block.
Now, let's consider the forces acting on the block as it travels down the track. The only force acting on the block is the force of friction. This force can be calculated using the equation:
Force of Friction = friction coefficient * normal force
The normal force is the force exerted by the surface on the block and is equal to the weight of the block, which is given by:
Normal Force = mass * gravity
Normal Force = 209 kg * 9.8 m/s^2 = 2046.2 N
Substituting this into the equation, we get:
-927.02 N = friction coefficient * 2046.2 N
Solving for the friction coefficient, we have:
friction coefficient = -927.02 N / 2046.2 N
≈ -0.453
The negative sign indicates that the friction force opposes the motion of the block.
Therefore, the coefficient of kinetic friction between the surface and the block over the 7 m track length is approximately 0.453.