A 297-kg motorcycle is accelerating up along a ramp that is inclined 30.0º above the horizontal. The propulsion force pushing the motorcycle up the ramp is 3560 N, and air resistance produces a force of 270 N that opposes the motion. Find the magnitude of the motorcycle's acceleration.
See Related Questions: Mon, 9-29-14, 12:56 AM.
To find the magnitude of the motorcycle's acceleration, we need to determine the net force acting on the motorcycle and then use Newton's second law, F = ma, where F is the net force and a is the acceleration.
First, let's break down the forces acting on the motorcycle along the incline. Since the motorcycle is accelerating up the ramp, the forces acting on it are as follows:
1. The force of gravity (Fg), which is acting vertically downward and can be calculated using the equation: Fg = mg, where m is the mass of the object (in this case, the motorcycle) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = (297 kg) * (9.8 m/s^2) = 2916.6 N
2. The propulsion force (Fp), which is pushing the motorcycle up the ramp.
Fp = 3560 N
3. The force of air resistance (Fr), which opposes the motion and acts in the opposite direction of the propulsion force.
Fr = 270 N
Now, let's find the net force acting on the motorcycle along the incline. Since the ramp is at an angle of 30.0º above the horizontal, we need to resolve the forces into their components.
1. The force of gravity can be resolved into two components: one parallel to the ramp (Fg_parallel) and one perpendicular to the ramp (Fg_perpendicular).
Fg_parallel = Fg * sin(30.0º) = 2916.6 N * sin(30.0º) = 1458.3 N
Fg_perpendicular = Fg * cos(30.0º) = 2916.6 N * cos(30.0º) = 2527.4 N
2. The propulsion force can be resolved into two components as well: one parallel to the ramp (Fp_parallel) and one perpendicular to the ramp (Fp_perpendicular).
Fp_parallel = Fp * cos(30.0º) = 3560 N * cos(30.0º) = 3083.2 N
Fp_perpendicular = Fp * sin(30.0º) = 3560 N * sin(30.0º) = 1780 N
The net force along the incline can be calculated by subtracting the forces opposing the motion from the forces propelling the motorcycle.
Net force along the incline = (Fp_parallel - Fr) - Fg_parallel
= (3083.2 N - 270 N) - 1458.3 N
= 2354.9 N - 1458.3 N
= 896.6 N
Finally, we can use Newton's second law, F = ma, to find the magnitude of the motorcycle's acceleration.
a = F_net / m
= 896.6 N / 297 kg
= 3.02 m/s^2
Therefore, the magnitude of the motorcycle's acceleration is 3.02 m/s^2.