Physics Question

Question

A contestant in a winter games event pushes a 43.0 kg block of ice across a frozen lake with a rope over his shoulder at a 25 degree angle. (Enter the magnitudes. Assume that friction is not negligible.)

(a) Calculate the minimum force F (in N) he must exert to get the block moving.

(b) What is its acceleration (in m/s²) once it starts to move, if that force is maintained?

Answer 1

m = 43.0 kg, θ = 25°

(a) The x-component of F must counter-balance the static frictional force fₛ in order to get the block moving.

The y-component of F impacts fₛ by decreasing the normal force on the block.

Fₓ = F cos θ and Fᵧ = F sin θ

Fₙ = mg – Fᵧ = mg – F sin θ

fₛ = μₛ Fₙ = μₛ( mg – F sin θ)

The minimum force needed to get the block moving is given by Fₓ > fₛ which means that

F cos θ > μₛ( mg – F sin θ) = μₛmg – μₛ F sin θ

F(cos θ + μₛ sin θ) > μₛmg

F > (μₛmg)/(cos θ + μₛ sin θ)

Plugging in numbers (m = 43.0 kg, θ = 25°, g = 9.8 m/s²)

F > (421.4μₛ)/(0.906 + 0.423μₛ)

where μₛ is coefficient of static friction

(b) F_net = F cos θ – μₖ( mg – F sin θ) = F(cos θ + μₖ sin θ) – μₖmg

a = F_net/m = F(cos θ + μₖ sin θ)/m – μₖg

Plugging in numbers (m = 43.0 kg, θ = 25°, g = 9.8 m/s²)

a = (0.0211 + 0.0098 μₖ)F – 9.8 μₖ

where μₖ is coefficient of kinetic friction