A 200kg wagon is pushed along a horizontal surface at a constant 10m/s when it is released. After rolling a distance of 45m the wagon comes to a complete stop. Find the coefficient of friction.
To find the coefficient of friction, we need to use the equation for kinetic friction force:
Frictional Force = μ * Normal force
In this case, the wagon is initially pushed along a horizontal surface with a constant velocity. This means that the force of kinetic friction is acting in the opposite direction to the motion of the wagon, eventually bringing it to a stop.
Since the wagon comes to a complete stop, the frictional force is equal to the force that was initially pushing the wagon forward. In other words, the frictional force is equal to the force of gravity acting on the wagon.
Force of Friction = Force of Gravity
The force of gravity (also known as weight) can be calculated using the equation:
Force of Gravity = mass * gravitational acceleration
In this case, the mass of the wagon is 200 kg, and the gravitational acceleration is approximately 9.8 m/s².
Force of Gravity = 200 kg * 9.8 m/s²
Now, we can substitute the force of gravity into the equation for kinetic friction:
Kinetic Frictional Force = μ * Normal force = Force of Gravity
Since the wagon is on a horizontal surface and there is no vertical acceleration, the normal force (N) is equal to the force of gravity:
Normal force (N) = Force of Gravity = 200 kg * 9.8 m/s²
Therefore, the equation for kinetic friction becomes:
Kinetic Frictional Force = μ * Force of Gravity = 200 kg * 9.8 m/s²
To solve for the coefficient of friction (μ), we need to rearrange the equation:
μ = (Kinetic Frictional Force) / (Force of Gravity)
However, we need the value of the kinetic frictional force to solve for μ. We can find it using the work-energy principle:
Work done by friction = Change in kinetic energy
The work done by friction can be calculated as:
Work done by friction = Force of Friction * Distance
In this case, the distance traveled by the wagon is given as 45 m.
Work done by friction = Kinetic Frictional Force * 45 m
Since the wagon comes to a complete stop, the change in kinetic energy is equal to the initial kinetic energy of the wagon:
Change in kinetic energy = 1/2 * mass * (final velocity)² - 1/2 * mass * (initial velocity)²
In this case, the initial velocity is 10 m/s and the final velocity is 0 m/s.
Change in kinetic energy = 0 - 1/2 * 200 kg * (10 m/s)²
Now, we can equate the work done by friction to the change in kinetic energy:
Kinetic Frictional Force * 45 m = 0 - 1/2 * 200 kg * (10 m/s)²
Simplifying the equation:
Kinetic Frictional Force = - 1/2 * 200 kg * (10 m/s)² / 45 m
Plugging this value into our equation for μ:
μ = Kinetic Frictional Force / Force of Gravity
μ = (- 1/2 * 200 kg * (10 m/s)² / 45 m) / (200 kg * 9.8 m/s²)
Calculating this expression will give us the coefficient of friction.