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Two blocks of masses #m_1# and #m_2# are connected with a light spring of force constant #K# and whole system is kept on a frictionless horizontal surface the masses are applied forces #f_1# and #f_2# respectively?

At any time the blocks have accelerations a. The value of spring force is?

Answer 1

Sebastian Acosta

See below.

If the acceleration is the same then

#m_1 vec a = vec f_1 + vec T# #m_2 vec a = vec f_2 - vec T#

adding both equations

#vec a = 1/(m_1+m_2)(vec f_1+vec f_2)#

and now subtracting

#vec T = 1/2(m_1-m_2)vec a -1/2(vec f_1-vec f_2) = #

#=1/2(m_1-m_2)/(m_1+m_2)(vec f_1+vec f_2)-1/2(vec f_1-vec f_2) #

#T = 1/2( (m_1 - m_2)/ (m_1 + m2)-1)vec f_1 + 1/2( (m_1 - m_2)/ (m_1 + m_2)+1)vec f_2#

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Answer 2

Nora Arzate

The system will undergo oscillatory motion due to the forces applied by ( f_1 ) and ( f_2 ) respectively, and the restoring force from the spring. The motion will depend on the masses ( m_1 ) and ( m_2 ), the force constants of the spring ( K ), and the magnitudes and directions of ( f_1 ) and ( f_2 ). The specific behavior can be analyzed using principles of classical mechanics and Newton's laws of motion.

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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.