A model train, with a mass of #12 kg#, is moving on a circular track with a radius of #9 m#. If the train's kinetic energy changes from #36 j# to #27 j#, by how much will the centripetal force applied by the tracks change by?
The change in centripetal force is
The centripetal force is
The kinetic energy is
The variation of kinetic energy is
The variation of centripetal force is
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The change in centripetal force applied by the tracks will be 9 N.
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The centripetal force ( F_c ) acting on an object moving in a circular path can be calculated using the formula:
[ F_c = \frac{mv^2}{r} ]
where ( m ) is the mass of the object, ( v ) is its velocity, and ( r ) is the radius of the circular path.
First, we need to find the initial velocity of the model train using its initial kinetic energy ( KE_1 ) and then the final velocity using its final kinetic energy ( KE_2 ). The kinetic energy ( KE ) is given by the formula:
[ KE = \frac{1}{2}mv^2 ]
Given that ( KE_1 = 36 , \text{J} ) and ( KE_2 = 27 , \text{J} ), we can set up the equations:
[ 36 = \frac{1}{2} \times 12 \times v_1^2 ] [ 27 = \frac{1}{2} \times 12 \times v_2^2 ]
Solving for ( v_1 ) and ( v_2 ):
[ v_1^2 = \frac{36 \times 2}{12} = 6 ] [ v_1 = \sqrt{6} \approx 2.45 , \text{m/s} ]
[ v_2^2 = \frac{27 \times 2}{12} = 4.5 ] [ v_2 = \sqrt{4.5} \approx 2.12 , \text{m/s} ]
Now, we can calculate the initial centripetal force ( F_{c1} ) and the final centripetal force ( F_{c2} ) using the formula ( F_c = \frac{mv^2}{r} ):
[ F_{c1} = \frac{12 \times (2.45)^2}{9} \approx 7.32 , \text{N} ] [ F_{c2} = \frac{12 \times (2.12)^2}{9} \approx 5.02 , \text{N} ]
The change in centripetal force is given by ( \Delta F_c = F_{c2} - F_{c1} ):
[ \Delta F_c = 5.02 , \text{N} - 7.32 , \text{N} = -2.3 , \text{N} ]
Therefore, the centripetal force applied by the tracks will decrease by approximately 2.3 N.
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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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