292 kg motorcycle is accelerating up a ramp that is 30 degrees above the horizontal.
Propulsion force pushing the motorcycle up ramp is 3150 N. Air resistance produces a force of 250 N that opposes the motion.
Find that motorcycle's acceleration.
To find the motorcycle's acceleration, we need to resolve the forces acting on it into their components along the ramp.
First, let's resolve the weight of the motorcycle. The weight can be broken down into two components: one parallel to the ramp and the other perpendicular to the ramp. The component parallel to the ramp will help in calculating the motorcycle's acceleration.
The weight (W) of the motorcycle can be calculated using the formula:
W = m * g,
where m is the mass of the motorcycle and g is the acceleration due to gravity (9.8 m/s²).
Given that the mass of the motorcycle (m) is 292 kg, the weight can be calculated as:
W = 292 kg * 9.8 m/s² = 2861.6 N.
Next, let's resolve the applied force and the air resistance force. The applied force is pushing the motorcycle up the ramp, while the air resistance force is opposing its motion. We need to break down these forces into their components parallel and perpendicular to the ramp.
The component of the applied force (Fa) parallel to the ramp is given by:
Fa_parallel = Fa * cos(θ),
where Fa is the propulsion force pushing the motorcycle up the ramp and θ is the angle of the ramp (30 degrees).
Given the applied force (Fa) is 3150 N and the angle of the ramp (θ) is 30 degrees, we can calculate the component of the applied force parallel to the ramp as:
Fa_parallel = 3150 N * cos(30°) = 2728.85 N.
The component of the air resistance force (Fr) parallel to the ramp is given by:
Fr_parallel = Fr * cos(180° - θ),
where Fr is the air resistance force opposing the motion.
Given the air resistance force (Fr) is 250 N, we can calculate the component of the air resistance force parallel to the ramp as:
Fr_parallel = 250 N * cos(180° - 30°) = 202.76 N.
Now, we can calculate the net force (F_net) acting on the motorcycle along the ramp by considering the forces acting parallel to the ramp:
F_net = Fa_parallel - Fr_parallel - W,
where W is the weight of the motorcycle.
Plugging in the values we calculated earlier:
F_net = 2728.85 N - 202.76 N - 2861.6 N = -335.51 N.
Since the net force (F_net) is in the negative direction along the ramp, the motorcycle is experiencing a deceleration. To get the magnitude of the acceleration (a), we can use Newton's second law of motion:
F_net = m * a,
where m is the mass of the motorcycle and a is the acceleration.
Rearranging the formula to solve for acceleration (a):
a = F_net / m.
Plugging in the values:
a = -335.51 N / 292 kg = -1.150 m/s².
Therefore, the magnitude of the motorcycle's acceleration is approximately 1.150 m/s².
Resolve all forces in the direction of the ramp, up=positive.
Propulsion force, P= +3150N
Resistance, A = -250N
due to weight of motocycle, W = -mgsinθ
Net force, F = P+A+W
Use F=ma to solve for a.