An object, previously at rest, slides #6 m# down a ramp, with an incline of #(3pi)/8 #, and then slides horizontally on the floor for another #8 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
The initial potential energy is
The total dissiped work is given by
so equating
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To find the kinetic friction coefficient, you can use the equation ( d = \frac{m v^2}{2 \mu g} ), where ( d ) is the distance, ( m ) is the mass of the object, ( v ) is the final velocity, ( \mu ) is the kinetic friction coefficient, and ( g ) is the acceleration due to gravity. Rearrange the equation to solve for ( \mu ):
[ \mu = \frac{m v^2}{2 d g} ]
First, calculate the final velocity of the object using the distance it travels on the ramp and the horizontal distance. Since the ramp and the floor are made of the same material, the final velocity on the ramp will be the same as the initial velocity on the floor.
For the distance traveled on the ramp, use the formula:
[ d = \frac{1}{2} a t^2 ]
Where ( a ) is the acceleration and ( t ) is the time. Since the object starts from rest, its initial velocity is 0. Therefore, the final velocity ( v ) on the ramp is the same as the final velocity ( v' ) on the floor.
Use the equation of motion to find ( v ):
[ v^2 = 2 a d ]
Substitute ( a = g \sin(\theta) ), where ( \theta = \frac{3\pi}{8} ), and solve for ( v ).
Once you have ( v ), you can calculate ( \mu ) using the rearranged equation. Make sure to use consistent units throughout the calculation.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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