A model train, with a mass of #3# #kg#, is moving on a circular track with a radius of #3# #m#. If the train's kinetic energy changes from #18# #J# to #0# #J#, by how much will the centripetal force applied by the tracks change by?

Answer 1

The centripetal force is given by #F=(mv^2)/r#. We are given the mass and the radius and can calculate the velocity from the kinetic energy. The change is #-12# #N#.

The centripetal force on an object of mass #m# #kg# in circular motion at velocity #v# #ms^-1# in a circle of radius #r# #m# is given by:
#F=(mv^2)/r#
In this case we know the mass is #3# #kg# and the radius is #3# #m#, but the velocity is not immediately obvious. We are given the kinetic energy at two moments in time, though, and we know that kinetic energy is:
#E_k=1/2mv^2#
We can rearrange this to make #v# the subject:
#v=sqrt((2E_k)/m)#
Since the kinetic energy at the second time is #0# #J#, the velocity is also #0# #ms^-1#, and therefore the centripetal force is also #0# #N#.
At the first time, the kinetic energy is #18# #J#, so:
#v=sqrt((2*18)/3)=sqrt(36/3)=sqrt12# #ms^-1#

Thus, the centripetal force is as follows:

#F=(mv^2)/r = (3*12)/3 = 12# #N# (since #(sqrt12)^2=12#)
So the centripetal force was #12# #N# at the beginning and #0# #N# at the end. Its change was therefore #-12# #N#.
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Answer 2

The change in kinetic energy can be calculated using the formula ΔKE = KE_final - KE_initial. ΔKE = 0 J - 18 J = -18 J. The centripetal force can be calculated using the formula F_c = (mv^2)/r, where m is the mass, v is the velocity, and r is the radius. Initial velocity can be calculated using the kinetic energy formula KE = 0.5 * m * v^2 rearranged to v = √(2 * KE / m) = √(2 * 18 J / 3 kg) = √12 m/s. Initial centripetal force can be calculated as F_c_initial = (3 kg * (√12 m/s)^2) / 3 m = 4 N. Final centripetal force can be calculated as F_c_final = (3 kg * 0 m/s^2) / 3 m = 0 N. The change in centripetal force is the difference between the initial and final centripetal forces: ΔF_c = F_c_final - F_c_initial = 0 N - 4 N = -4 N. Therefore, the centripetal force applied by the tracks will change by -4 N.

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Answer from HIX Tutor

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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