A sphere is attached to the ceiling of an elevator by a string. A second sphere

Question

A sphere is attached to the ceiling of an elevator by a string. A second sphere is attached to the first one by a second string. Both strings are of negligible mass. Here m_1 = m_2 = m = 2.99 kg. The elevator starts from rest and accelerates downward with a =1.35 m/s^2. What are the tensions in the two strings? If the elevator starts from rest and accelerates upward with the same acceleration, what will be the tension in the two strings? The maximum tension the two strings can withstand is 93.4 N. What maximum upward acceleration can the elevator have without having one of the strings break?

Explanation / Answer

Apply Newton's second law to the total force on the top sphere ( taking 'up' as positive,)

T1 - T2 - mg = ma

For the second (lowest) sphere, the net force

T2 - m g = m a

Therefore,

T2 = mg + ma and

T1 = 2mg + 2ma

With m = 2.99 kg and a= -1.35 m/s^2

T1 = 2( 2.99 kg * (9.81m/s^2 - 1.35m/s^2)) = 50.6 N

and

T2 = 25.3 N

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If a= +1.35 m/s^2 this gives

T1 = 2( 2.99kg * (9.81m/s^2 + 1.35m/s^2)) = 66.7 N

T2 = 33.4 N

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From the equation

T1 = 2(mg + ma)

2(mg + ma) < 93.4 N

a < 93.4 N /(2*2.99kg) + 9.81m/s^2

a < 5.92 m/s^2

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