Question of Physics

Question

A particle of mass m=1kg has position vector r

(t) given by r(t) = Ati+Bt²j+Ct³k (meters) where A;B, and C are constants. At t=1 second,

find

a) the torque,r, with respect to the origin, and

b) the angular momentum, L, with respect to the origin.

Answer 1

I assume that denoting the torque as "r" is just an unfortunate mistake,

and I will use "r" only for the position vector.

If I am wrong, and the problem statement is meant to indicate some intended

coincidence between the values of the position vector and the torque,

then ignore what I am writing.

We will need both the velocity and acceleration of the object.

Those are the first and second derivatives of r(t)

r(t) = Ati + Bt²j + Ct³k

r'(t) = Ai + 2Btj + 3Ct²k

r"(t) = 2B + 6Ctk

At time t=1

r(1) = Ai + Bj + Ck

r'(1) = Ai + 2Bj + 3Ck

r"(1) = 2Bj + 6Ck

a) The torque is defined as

r x F = r x mr"

Using the formula for the cross product at time t=1 and with m=1kg

r(1) x r"(1) = (6BC - 2BC)i + (0 - 6AC)j + (2AB -0)k = 4BCi - 6ACj + 2ABk

b) The angular momentum is defined as

r x p = r x mr'

Using the formula for the cross product at time t=1 and with m=1kg

r(1) x r'(1) = (2BC - 3BC)i + (AC - 3AC)j + (2AB - AB)k = -BCi -2ACj +ABk