Can an object have angular momentum without torque?

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Yes, an object can have angular momentum without torque. The physics behind it is that if no external force act on an object, its angular momentum is constant. That is based on the law of conservation of momentum. 

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Angular impulse = Change in angular momentum

[J] = [mvr]

[J] = [M1L2T–1]

Position vector about O (r_o) = 5i + 5√3j

Force vector (F) = 4i - 3j

Torque about O (τ_o) = r_o × F
τ_o = (5i + 5√3j) × (4i - 3j)
τ_o = -15k - 20√3k = (-15 - 20√3)k

Position vector about Q (r_q) = -5i + 5√3j

Torque about Q (τ_q) = r_q × F
τ_q = (-5i + 5√3j) × (4i - 3j)
τ_q = 15k - 20√3k = (15 - 20√3)k

r? = 10αt²î + 5β (t-5)?
v? = dr? /dt = 20αtî + 5β?
As L? = m (r? × v? )
So, at t=0, L=0
given L is same at t=t as at t=0
⇒ r? × v? = 0
⇒ (10αt²î + 5β (t-5)? ) × (20αtî + 5β? ) = 0
⇒ 50αβt² (î×? ) + 100αβt (t-5) (? ×î) = 0
⇒ 50αβt² k? - 100αβt (t-5) k? = 0
⇒ 50t² - 100t (t-5) = 0
⇒ 50t² - 100t² + 500t = 0
⇒ -

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