Final answer:
To lift a 100.0-kg crate up a 30° incline with an acceleration of 2.0 m/s² on a frictionless plane, a total force of 690 N is required. The calculation involves summing the gravitational force component acting down the slope and the force needed to produce the desired acceleration.
Step-by-step explanation:
Calculating the Required Force to Lift a Crate
The question involves finding the force required to lift a crate on an inclined plane with a specific acceleration, which is a classic example of applying Newton's second law of motion in the context of an inclined plane problem. To calculate this force, we need to account for both the gravitational component acting down the incline and the desired upward acceleration.
First, we determine the component of the gravitational force acting down the slope using the formula mg × sin(30°), where m is the mass of the crate and g is the acceleration due to gravity. For a 100.0-kg crate, this becomes 100.0 kg × 9.8 m/s² × sin(30°), which equals 490 N. Then we use Newton's second law, F = ma, to find the additional force required to achieve the 2.0 m/s² acceleration. This gives us 100.0 kg × 2.0 m/s² = 200 N. Adding these together, the total force required is 490 N + 200 N = 690 N.
Remember that force is a vector, and the forces acting along the incline must be added algebraically.