Assume a cyclist of weight mg can exert a force on the pedals equal to 0.90 mg on the average. If the pedals rotate in a circle of radius 18 cm, the wheels have a radius of 34 cm, and the front and back sprockets on which the chain runs have 42 and 19 teeth respectively (Fig. 7-32), determine the maximum steepness of hill the cyclist can climb at constant speeds. Assume the mass of the bike is 18 kg and that of the rider is 70 kg. Ignore friction.
Well, to determine the maximum hill steepness, we first need to figure out the maximum force the cyclist can exert on the pedals. Since the cyclist can exert a force equal to 0.90 mg on the average, the maximum force F_max would be 0.90 times the weight of the cyclist, or 0.90 times (mg).
Now let's calculate the weight of the cyclist. Since the mass of the cyclist is 70 kg, we can find the weight using the formula W = mg. Assuming the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the cyclist would be 70 kg times 9.8 m/s^2.
So, the weight of the cyclist is 686 N (Newtons).
Now, let's calculate the maximum force F_max that the cyclist can exert. F_max = 0.90 times (686 N).
F_max = 0.90 times (686 N) = 617.4 N (approximately).
Considering the gears on the bicycle, we need to calculate the gear ratio. The gear ratio is calculated by dividing the number of teeth on the front sprocket by the number of teeth on the back sprocket. In this case, the front sprocket has 42 teeth and the back sprocket has 19 teeth. So, the gear ratio R would be 42/19.
Now, to find the force applied to the wheels, we multiply the maximum force F_max by the gear ratio R. So, the force F_wheels would be 617.4 N times the gear ratio (42/19).
F_wheels = 617.4 N times (42/19) = 1369.93 N (approximately).
The next step is to calculate the torque exerted by the force F_wheels on the wheels. The torque is calculated by multiplying the force by the radius. In this case, the radius of the wheels is 34 cm, or 0.34 m. So, the torque T_wheels would be the force F_wheels times the radius 0.34 m.
T_wheels = 1369.93 N times 0.34 m = 465.57 Nm (approximately).
Now, let's calculate the torque exerted by the cyclist's force on the pedals. The torque T_pedals is calculated by multiplying the force F_max by the radius of the pedals. In this case, the radius of the pedals is 18 cm, or 0.18 m.
T_pedals = F_max times 0.18 m = 111.132 Nm (approximately).
To determine the maximum steepness of the hill that the cyclist can climb at constant speed, we need to compare the torque exerted by the cyclist's force on the pedals (T_pedals) to the torque exerted by the force on the wheels (T_wheels). If T_pedals is greater than T_wheels, then the cyclist can climb the hill at constant speed.
So, the maximum steepness of the hill the cyclist can climb is determined by the equation T_pedals >= T_wheels.
111.132 Nm >= 465.57 Nm.
Well, it seems that T_pedals is significantly smaller than T_wheels. Therefore, the poor cyclist won't be able to conquer hills with such steepness and will have to find less challenging paths. Remember, it's all about torque, not about torturing yourself!
To determine the maximum steepness of the hill the cyclist can climb at a constant speed, we need to consider the forces acting on the cyclist.
First, let's calculate the force the cyclist can exert on the pedals. The problem states that the force the cyclist can exert is 0.90 times their weight (mg).
Given that the mass of the rider is 70 kg, the force the cyclist can exert on the pedals is:
Force on pedals = 0.90 * (mass of rider * g) = 0.90 * (70 kg * 9.8 m/s^2) = 617.4 N
Next, let's consider the forces acting on the cyclist while climbing the hill.
1. Weight force: The weight of the cyclist (rider + bike) acts downward and is given by:
Weight force = (mass of rider + mass of bike) * g
Weight force = (70 kg + 18 kg) * 9.8 m/s^2
Weight force = 784.8 N
2. Normal force: Since the cyclist is on an incline, there is a perpendicular force called the normal force, which is equal in magnitude and opposite in direction to the weight force.
The normal force is equal to the weight force: Normal force = 784.8 N
3. Force due to pedaling: The force exerted on the pedals generates a rotational force on the rear wheel. The radius of the rear wheel is given as 34 cm, which is equal to 0.34 m. The force generated can be calculated using the formula:
Force on rear wheel = (Force on pedals * radius of rear wheel) / radius of pedal
Given that the radius of the pedal is 18 cm, which is equal to 0.18 m, and the Force on pedals is 617.4 N, we can calculate the force on the rear wheel:
Force on rear wheel = (617.4 N * 0.34 m) / 0.18 m = 1152.8 N
Now, let's consider the forces acting parallel and perpendicular to the direction of motion.
- Force parallel to the incline: The force parallel to the incline is the component of the force on the rear wheel acting along the incline. This force helps the cyclist move up the hill. The force parallel to the incline is given by:
Force parallel = Force on rear wheel * sin(θ)
where θ is the angle of inclination of the hill.
- Force perpendicular to the incline: The force perpendicular to the incline is the component of the force on the rear wheel acting perpendicular to the incline. This force is responsible for balancing the weight force and normal force, and it does not contribute to the motion. The force perpendicular to the incline is given by:
Force perpendicular = Force on rear wheel * cos(θ)
Since the cyclist is moving at a constant speed, the force parallel to the incline must balance the gravitational and frictional forces. Therefore, the force parallel to the incline is equal to the force of gravity acting down the incline plus the force of friction acting up the incline.
Force parallel = Force of gravity + Force of friction
The force of gravity down the incline is given by:
Force of gravity = Weight force * sin(θ)
The force of friction opposing the motion up the incline is given by:
Force of friction = Weight force * cos(θ)
Therefore, we have the equation:
Force parallel = Weight force * sin(θ) + Weight force * cos(θ)
Substituting the values, we can solve for θ (the maximum steepness of the hill the cyclist can climb at constant speed):
Force on rear wheel * sin(θ) + Force on rear wheel * cos(θ) = Weight force * sin(θ) + Weight force * cos(θ)
1152.8 N * sin(θ) + 1152.8 N * cos(θ) = 784.8 N * sin(θ) + 784.8 N * cos(θ)
Now, you can solve this equation for θ using algebraic manipulation and trigonometric identities.