An object, previously at rest, slides #7 m# down a ramp, with an incline of #(pi)/3 #, and then slides horizontally on the floor for another #4 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
The coefficient of kinetic friction is
At the top of the ramp, the object possesses potential enegy.
At the botton of the ramp, part of the potential energy is converted into kinetic energy and the work done by the frictional force.
Therefore,
so,
On the horizontal part,
The deceleration is calculated with Newton's Second Law
We apply the equation of motion
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To find the kinetic friction coefficient (( \mu )) of the material, you need to analyze the motion of the object.
 Calculate the gravitational force component parallel to the ramp's surface. [ F_{\text{parallel}} = m \times g \times \sin(\theta) ] Where:
 ( m ) is the mass of the object.
 ( g ) is the acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 )).
 ( \theta ) is the angle of inclination of the ramp (( \frac{\pi}{3} )).
 Calculate the work done by gravity on the ramp. [ W_{\text{ramp}} = F_{\text{parallel}} \times d_{\text{ramp}} ] Where:
 ( d_{\text{ramp}} ) is the distance along the ramp (7 m).
 Calculate the work done by the friction force on the horizontal surface. [ W_{\text{friction}} = \mu \times m \times g \times d_{\text{floor}} ] Where:
 ( d_{\text{floor}} ) is the distance along the horizontal surface (4 m).

Since the object starts from rest, the work done by gravity on the ramp is equal to the work done by the friction force on the horizontal surface (by the workenergy principle). [ W_{\text{ramp}} = W_{\text{friction}} ]

Set the expressions for ( W_{\text{ramp}} ) and ( W_{\text{friction}} ) equal to each other and solve for ( \mu ).
[ \mu = \frac{F_{\text{parallel}} \times d_{\text{ramp}}}{m \times g \times d_{\text{floor}}} ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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 If the man pushes with a force of 1005N, the box will slide up the ramp at a rate of 1.1m/s². How much normal force is acting on the box?
 An object with a mass of #2 kg# is on a plane with an incline of #pi/8 #. If the object is being pushed up the plane with # 3 N # of force, what is the net force on the object?
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