# A model train, with a mass of #8 kg#, is moving on a circular track with a radius of #2 m#. If the train's kinetic energy changes from #72 j# to #36 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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To find the change in centripetal force, we'll first calculate the initial and final speeds of the train using the given kinetic energies. Then, we'll use the formula for centripetal force to determine the initial and final forces, and finally, find the difference.

The formula for kinetic energy is ( KE = \frac{1}{2} mv^2 ). Given that the initial kinetic energy (( KE_1 )) is 72 J and the mass (( m )) is 8 kg, we can find the initial speed (( v_1 )).

( 72 = \frac{1}{2} \times 8 \times (v_1)^2 ) ( v_1 = \sqrt{\frac{2 \times 72}{8}} ) ( v_1 = \sqrt{18} ) ( v_1 = 4.24 , m/s )

Similarly, for the final kinetic energy (( KE_2 )) of 36 J, we can find the final speed (( v_2 )).

( 36 = \frac{1}{2} \times 8 \times (v_2)^2 ) ( v_2 = \sqrt{\frac{2 \times 36}{8}} ) ( v_2 = \sqrt{9} ) ( v_2 = 3 , m/s )

Now, we can find the initial and final centripetal forces (( F_1 ) and ( F_2 )) using the formula ( F = \frac{mv^2}{r} ).

For the initial force (( F_1 )): ( F_1 = \frac{8 \times (4.24)^2}{2} ) ( F_1 = \frac{8 \times 18}{2} ) ( F_1 = \frac{144}{2} ) ( F_1 = 72 , N )

For the final force (( F_2 )): ( F_2 = \frac{8 \times (3)^2}{2} ) ( F_2 = \frac{8 \times 9}{2} ) ( F_2 = \frac{72}{2} ) ( F_2 = 36 , N )

The change in centripetal force (( \Delta F )) is the difference between the initial and final forces: ( \Delta F = F_2 - F_1 ) ( \Delta F = 36 , N - 72 , N ) ( \Delta F = -36 , N )

Therefore, the centripetal force applied by the tracks will change by -36 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.

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