A box with an initial speed of #6 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #5/3 # and an incline of #(3 pi )/8 #. How far along the ramp will the box go?
The distance is
Consequently, the object's net force is
Newton's Second Law states
So
A deceleration is indicated by the negative sign.
We utilize the equation of motion.
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To find the distance along the ramp the box will go, you can use the equation:
[ d = \frac{{v_0^2}}{{2g}} \times \left( \frac{{\sin(2\theta)}}{{\mu_k + \cos(\theta)}} \right) ]
where:
- ( d ) is the distance along the ramp,
- ( v_0 ) is the initial speed (6 m/s),
- ( g ) is the acceleration due to gravity (9.8 m/s²),
- ( \theta ) is the incline angle ((3π)/8),
- ( \mu_k ) is the kinetic friction coefficient (5/3).
Plug in the values and solve for ( d ):
[ d = \frac{{6^2}}{{2 \times 9.8}} \times \left( \frac{{\sin(2 \times (3\pi)/8)}}{{5/3 + \cos((3\pi)/8)}} \right) ]
[ d ≈ 2.38 , \text{m} ]
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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.
- An object with a mass of #8 kg# is lying still on a surface and is compressing a horizontal spring by #5/4 m#. If the spring's constant is #3 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?
- An object with a mass of #5 kg# is on a surface with a kinetic friction coefficient of # 4 #. How much force is necessary to accelerate the object horizontally at #1 m/s^2#?
- An object, previously at rest, slides #3 m# down a ramp, with an incline of #(3pi)/8 #, and then slides horizontally on the floor for another #2 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
- Horizontal force of 50N accelerates a block of mass 3.0 kg up an incline with an A=2.0ms^-2. The angle with horizontal is 30^o. Find a) force of friction between the box and the incline b) the coefficient of kinetic friction between box and the incline?
- An object with a mass of # 18 kg# is lying on a surface and is compressing a horizontal spring by #40 cm#. If the spring's constant is # 4 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?

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