An ice skater with a mass of 60.0kg is moving at 20.0 m/s coasts to a halt in 2.00×10^2m on a smooth ice surface. What is the coefficient of friction between the ice and the skates?
KE of skater=work done by ice
1/2(60)20^2=mu*60*9.8*200
solve for mu
To find the coefficient of friction between the ice and the skates, we can use the concept of work-energy principle. The work-energy principle states that the work done on an object is equal to its change in kinetic energy. In this case, the work done on the ice skater comes from the friction between the skates and the ice surface.
The work done by friction can be calculated using the formula:
Work = Force × distance
The frictional force can be calculated using the formula:
Force = μ × Normal force
Here, μ is the coefficient of friction and Normal force is the force exerted by the ice surface on the skater in the vertical direction. Since the skater is moving horizontally, the vertical Normal force is equal to the weight of the skater, which is given by:
Normal force = Mass × Gravitational acceleration
Gravitational acceleration is approximately 9.8 m/s^2.
Let's calculate the work done by friction:
Work = Force × distance
Using the formula Force = μ × Normal force, we have:
Work = (μ × Mass × Gravitational acceleration) × distance
The work done on the skater is equal to the change in kinetic energy, which can be calculated as:
Change in kinetic energy = (1/2) × Mass × (final velocity)^2 - (1/2) × Mass × (initial velocity)^2
The initial velocity is given as 20.0 m/s and the final velocity is 0 m/s (since the skater comes to a halt).
Setting the work done by friction equal to the change in kinetic energy, we can solve for the coefficient of friction (μ). Rearranging the equation, we get:
(μ × Mass × Gravitational acceleration) × distance = (1/2) × Mass × (final velocity)^2 - (1/2) × Mass × (initial velocity)^2
Simplifying further:
μ × Gravitational acceleration × distance = (1/2) × (final velocity)^2 - (1/2) × (initial velocity)^2
Finally, we can solve for the coefficient of friction (μ) by rearranging the equation:
μ = [(1/2) × ((final velocity)^2 - (initial velocity)^2)] / (Gravitational acceleration × distance)
Plugging in the given values, we can calculate the coefficient of friction between the ice and the skates.