Radial and transverse force on block B

Data

m = 1000 g = 1 kg; r = 2(1 + cos 2πt) m; θ = 2πt rad. At t = 0.8 s: 2πt = 5.0265 rad.

• cos 2πt = 0.3090, sin 2πt = −0.9511
• r = 2(1.3090) = 2.618 m
• ṙ = −4π sin 2πt = −4π(−0.9511) = 11.95 m/s
• r̈ = −8π² cos 2πt = −78.96(0.3090) = −24.40 m/s²
• θ̇ = 2π = 6.283 rad/s, θ̈ = 0

Acceleration components

• a_r = r̈ − rθ̇² = −24.40 − 2.618(39.48) = −127.75 m/s²
• a_θ = rθ̈ + 2ṙθ̇ = 0 + 2(11.95)(6.283) = 150.18 m/s²

Force components (horizontal plane, so no weight effect)

• F_r = m a_r = 1 × (−127.75) = −127.75 N (directed toward O)
• F_θ = m a_θ = 1 × 150.18 = 150.18 N (transverse, in the direction of increasing θ)

Resultant force = √(127.75² + 150.18²) = 197.2 N.

Principle of impulse and momentum

Starting from Newton's second law, F = m dv/dt, multiply by dt and integrate between times t₁ and t₂:

∫ F dt = m v₂ − m v₁

• Linear impulse = ∫ F dt (units N·s): the effect of a force acting over a time interval.
• Linear momentum = m v (units kg·m/s).
• The principle states: the linear impulse of the resultant force on a particle during a time interval equals the change in its linear momentum. In vector form m v₁ + ∫ F dt = m v₂, with separate component equations along x, y, z.
• If the resultant force is zero, momentum is conserved. The principle is most useful when the force is given as a function of time or when velocity and time (not displacement) are involved, for example impact.