Final answer:
To calculate the force that must be applied by the bicycle chain, one needs to consider the resistive force, wheel's mass, and desired acceleration, using physics concepts of torque, inertia, and angular acceleration.
Step-by-step explanation:
Finding the Force on a Bicycle Chain
To determine the force that must be applied by a chain passing over a sprocket to give a bicycle wheel an acceleration of 4.55 rad/s² when a resistive force of 123 N is applied tangent to the rim, we need to use the concepts of torque, inertia, and angular acceleration from physics. The torque (τ) caused by the resistive force is τ = r × F, where 'r' is the radius of the wheel and 'F' is the applied force. The wheel's moment of inertia (I) for a hoop is I = mr², where 'm' is the mass of the wheel and 'r' is the radius. The angular acceleration (α) is related to the torque and moment of inertia by the equation τ = Iα.
Given:
- Diameter of wheel = 64.2 cm (Radius = 32.1 cm)
- Mass of wheel = 1.73 kg
- Resistive force = 123 N
- Desired angular acceleration = 4.55 rad/s²
- Diameter of sprocket = 8.95 cm (Radius = 4.475 cm)
First, we calculate the torque needed to overcome the resistive force:
τ = 32.1 cm × 123 N = 3949.3 N·cm
Then, we calculate the wheel's moment of inertia:
I = 1.73 kg × (32.1 cm)²
Next, we calculate the torque required for the desired angular acceleration:
τ' = Iα
Finally, we use the torque and the radius of the sprocket to find the force the chain must exert:
F' = τ' / (4.475 cm)
By solving all these steps, we can find the force required on the bicycle chain.