A model train with a mass of #3 kg# is moving along a track at #15 (cm)/s#. If the curvature of the track changes from a radius of #35 cm# to #12 cm#, by how much must the centripetal force applied by the tracks change?
The change in centripetal force is
The centripetal force is
The variation in centripetal force is
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The centripetal force required for circular motion is given by the formula:
[ F_c = \frac{mv^2}{r} ]
Where:
- ( F_c ) is the centripetal force.
- ( m ) is the mass of the object.
- ( v ) is the velocity of the object.
- ( r ) is the radius of the circular path.
Given:
- ( m = 3 ) kg
- ( v = 15 ) cm/s
- Initial radius ( r_1 = 35 ) cm
- Final radius ( r_2 = 12 ) cm
Calculate the initial centripetal force:
[ F_{c1} = \frac{m \cdot v^2}{r_1} ]
[ F_{c1} = \frac{3 \cdot (15)^2}{35} ]
[ F_{c1} \approx 19.29 \text{ N} ]
Calculate the final centripetal force:
[ F_{c2} = \frac{m \cdot v^2}{r_2} ]
[ F_{c2} = \frac{3 \cdot (15)^2}{12} ]
[ F_{c2} = 56.25 \text{ N} ]
The change in centripetal force is:
[ \Delta F_c = F_{c2} - F_{c1} ]
[ \Delta F_c = 56.25 - 19.29 ]
[ \Delta F_c \approx 36.96 \text{ N} ]
So, the centripetal force applied by the tracks must increase by approximately 36.96 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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- A model train with a mass of #1 kg# is moving along a track at #6 (cm)/s#. If the curvature of the track changes from a radius of #8 cm# to #9 cm#, by how much must the centripetal force applied by the tracks change?
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