Classical Mechanics - Torque

A solid cone hangs from a frictionless pivot at the origin O as shown above. If î, ĵ and k̂ are unit vectors, and a, b, and c are positive constants, which of the following forces F applied to the rim of the cone at point P results in a torque τ on the cone with a negative component τ_z?

A. F = ak̂, P is (0, b, −c)
B. F = −ak̂, P is (0, −b, −c)
C. F = aĵ, P is (–b, 0, −c)
D. F = aĵ, P is (b, 0, −c)
E. F = −ak̂, P is (−b, 0, −c)

(GR9277 #08)

Solution:

Torque:

Result desired: negative component of τ_z so we are looking for minus k̂ component.

For k̂ component: r_xF_y − r_yF_x

Thus, F_x and F_y cannot be zero

(A) FALSE
 F = ak̂ → F_x = 0 and F_y = 0

(B) and (E) FALSE
F = −ak̂ → F_x = 0 and F_y = 0

(C) TRUE
F = aĵ → F_x = 0, F_y = a,  r_x = −b, r_y = 0,
P is (–b, 0, −c) → r_x = −b, r_y = 0
r_xF_y − r_yF_x = −ab − 0 =  −abk̂ (negative k̂ component)

(D) FALSE
F = aĵ → F_x = 0, F_y = a,
P is (b, 0, −c) → r_x = b, r_y = 0,
→ r_xF_y − r_yF_x = ab − 0 =  abk̂ (positive k̂ component)

Answer: C