A box of mass 27 kg is in the bed of a truck on a hill that is inclined at an angle of 26.1° above the horizontal. The tailgate of the truck is open. The truck is currently accelerating up the hill with a magnitude of 1.67 m/s2
The truck is now at rest on the hill. If the driver guns the engine so that the truck has an acceleration of 2.81 m/s2 up the hill, then find the acceleration of the box with respect to the (essentially) inertial frame of the earth. Use 0.24 as the coefficient of kinetic friction between the box and the truck.
To find the acceleration of the box with respect to the inertial frame of the Earth, we need to consider the forces acting on the box.
1. Calculate the weight force: The weight force (mg) acts vertically downward, always perpendicular to the inclined plane. The weight force can be calculated using the formula weight = mass × gravity, where mass (m) is given as 27 kg, and gravity (g) is approximately 9.8 m/s^2.
weight = 27 kg × 9.8 m/s^2 = 264.6 N (rounded to one decimal place)
2. Decompose the weight force: The weight force can be decomposed into two components: one parallel to the inclined plane (mg sinθ) and one perpendicular to the inclined plane (mg cosθ). Here, θ represents the angle of the hill, which is 26.1°.
Parallel component: weight_parallel = weight × sinθ
weight_parallel = 264.6 N × sin(26.1°) = 113.0 N (rounded to one decimal place)
Perpendicular component: weight_perpendicular = weight × cosθ
weight_perpendicular = 264.6 N × cos(26.1°) = 235.6 N (rounded to one decimal place)
3. Calculate the frictional force: The frictional force (f_friction) opposes the motion and acts parallel to the inclined plane. The frictional force can be determined using the formula friction = coefficient of friction × normal force. The coefficient of kinetic friction (μ_k) is given as 0.24, and the normal force is equal to the perpendicular component of the weight force.
friction = μ_k × weight_perpendicular
friction = 0.24 × 235.6 N = 56.5 N (rounded to one decimal place)
4. Determine the net force: The net force (F_net) acting on the box is the difference between the parallel component of the weight force and the frictional force, since the frictional force opposes the motion.
F_net = weight_parallel - friction
F_net = 113.0 N - 56.5 N = 56.5 N
5. Calculate the acceleration: To find the acceleration of the box, we can use Newton's second law, which states that acceleration is equal to the net force divided by the mass of the object.
acceleration = F_net / mass
acceleration = 56.5 N / 27 kg = 2.09 m/s^2 (rounded to two decimal places)
Therefore, the acceleration of the box with respect to the inertial frame of the Earth is approximately 2.09 m/s^2.