Answer :
Final answer:
The force needed to accelerate a 100-kg crate up a frictionless 30° incline at 2.0 m/s² is approximately 690.5 N, which rounds to 700 N and closely matches option A, 686 N.
Explanation:
To determine the force needed to accelerate a 100.0-kg crate up a frictionless inclined plane, we must analyze the forces acting parallel and perpendicular to the surface of the incline. The only force acting parallel to the incline that causes acceleration is the component of the gravitational force, plus the force we apply to achieve an acceleration of 2.0 m/s².
First, we calculate the gravitational force component parallel to the incline using the formula Fg parallel = m * g * sin(θ), where m is the mass, g is the gravitational acceleration (9.81 m/s²), and θ is the incline angle. So, Fg parallel = 100.0 kg * 9.81 m/s² * sin(30°) = 490.5 N. This force acts downhill. To move the crate uphill, the applied force must overcome this downhill force.
Next, we can use Newton's second law, F = m*a, to determine the force needed to accelerate the crate up the hill. Total force needed (Ftotal) is the sum of the force to overcome gravity plus the force to accelerate the crate: Ftotal = Fg parallel + Fapplied. Since the acceleration is 2.0 m/s², we calculate Fapplied as Fapplied = m*a = 100.0 kg * 2.0 m/s² = 200.0 N. Therefore, Ftotal = 490.5 N + 200.0 N = 690.5 N. However, this is not an option given in the question, and rounding it to one significant figure as the options suggests, we obtain an approximate value of 700 N, which is closest to option A.
