A 297-kg motorcycle is accelerating up along a ramp that is inclined 25.9° above the horizontal. The propulsion force pushing the motorcycle up the ramp is 3156 N, and air resistance produces a force of 300 N that opposes the motion. Find the magnitude of the motorcycle's acceleration.
gravity also opposes the motion with a force of __ g * 297 * sin(25.9º)
sum the forces; then find the acceleration with __ f = m a
To find the magnitude of the motorcycle's acceleration, we need to apply Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this scenario, the net force is the difference between the propulsion force pushing the motorcycle up the ramp and the force of air resistance opposing the motion. The net force is given by:
Net force = Propulsion force - Force of air resistance
Net force = 3156 N - 300 N = 2856 N
Now, we can use the resolved forces along the ramp and perpendicular to the ramp to calculate the acceleration.
The component of the weight of the motorcycle perpendicular to the ramp is:
Weight perpendicular to ramp = m * g * cos(θ)
where m is the mass of the motorcycle (297 kg), g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of inclination of the ramp (25.9°).
Weight perpendicular to ramp = 297 kg * 9.8 m/s² * cos(25.9°) = 2566.7 N
The component of the propulsion force along the ramp is:
Propulsion force along ramp = Propulsion force * sin(θ)
Propulsion force along ramp = 3156 N * sin(25.9°) = 1363.8 N
Now, we can find the net force along the ramp by subtracting the force due to weight perpendicular to the ramp:
Net force along ramp = Propulsion force along ramp - Weight perpendicular to ramp
Net force along ramp = 1363.8 N - 2566.7 N = -1202.9 N (the negative sign indicates that the net force is in the opposite direction of the motion)
Finally, we can calculate the magnitude of the motorcycle's acceleration using Newton's second law:
Net force along ramp = mass * acceleration
-1202.9 N = 297 kg * acceleration
Solving for acceleration, we get:
acceleration = -1202.9 N / 297 kg ≈ -4.05 m/s²
Since acceleration cannot be negative in this context, we take the absolute value:
Magnitude of acceleration = 4.05 m/s²
Therefore, the magnitude of the motorcycle's acceleration is approximately 4.05 m/s².