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The position vector of a particle of mass m = 0.1 kg is given by r(t) = (10/3)*t³ * i_hat + 5*t² * j_hat (in metres), where t is in seconds. At t = 1 s, which of the following statements are correct? (A) The velocity is v = (10*i_hat + 10*j_hat) m/s (B) The angular momentum about the origin is L = -(5/3)*k_hat N*m*s (C) The net force on the particle is F = (i_hat + 2*j_hat) N (D) The torque about the origin is tau = -(20/3)*k_hat N*m

1. The velocity is v = (10*i_hat + 10*j_hat) m/s
2. The angular momentum about the origin is L = -(5/3)*k_hat N*m*s
3. The net force on the particle is F = (i_hat + 2*j_hat) N
4. The torque about the origin is tau = -(20/3)*k_hat N*m

Correct answer: The velocity is v = (10*i_hat + 10*j_hat) m/s

Solution

At t=1: v = 10*i_hat + 10*j_hat (A correct). r = (10/3)*i_hat + 5*j_hat. L = m*(r x v) = 0.1*[(10/3)*i_hat + 5*j_hat] x [10*i_hat + 10*j_hat] = 0.1*(100/3 - 50)*k_hat = 0.1*(-50/3)*k_hat = -(5/3)*k_hat (B correct). a = 20*i_hat + 10*j_hat → F = 0.1*(20*i_hat + 10*j_hat) = 2*i_hat + j_hat N (C wrong, F has swapped components). tau = r x F = [(10/3)*i_hat + 5*j_hat] x [2*i_hat + j_hat] = (10/3 - 10)*k_hat = -(20/3)*k_hat (D correct).

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