A bicyclist of mass 65 kg stands on pedal. The crank is .170 m long and makes a 45 angle with the vertical. The crank is attached to the chain wheel which is 9.70 cm. What force must the chain exert to keep the wheel from turning?
The force required to keep the wheel from turning is equal to the torque applied by the bicyclist's weight multiplied by the radius of the chain wheel.
Torque = 65 kg * 9.7 cm * sin(45°) = 463.5 Nm
Force = Torque / Radius = 463.5 Nm / 9.7 cm = 47.8 N
To find the force the chain must exert to keep the wheel from turning, we need to consider the forces acting on the system.
The weight of the bicyclist can be calculated as the force due to gravity acting on the mass:
Weight = mass × acceleration due to gravity
= 65 kg × 9.8 m/s²
= 637 N
Next, we need to resolve the weight force into its horizontal and vertical components.
The vertical component of the weight force can be calculated as the weight multiplied by the sine of the angle between the crank and the vertical:
Vertical component = Weight × sin(angle)
= 637 N × sin(45°)
≈ 450 N
The horizontal component of the weight force can be calculated as the weight multiplied by the cosine of the angle between the crank and the vertical:
Horizontal component = Weight × cos(angle)
= 637 N × cos(45°)
≈ 451 N
Now let's consider the forces acting on the crank. We have the vertical component of the weight force acting downwards and the force exerted by the chain acting upwards. These forces form a force couple, which means they must be equal in magnitude but opposite in direction to maintain equilibrium.
So, the magnitude of the force exerted by the chain is equal to the vertical component of the weight force:
Force by chain = Vertical component of weight
≈ 450 N
Therefore, the force the chain must exert to keep the wheel from turning is approximately 450 N.
To find the force that the chain must exert to keep the wheel from turning, we need to consider the equilibrium of forces acting on the bicyclist.
First, let's resolve the weight of the bicyclist into two components:
1. Vertical component: mg * cos(45°)
2. Horizontal component: mg * sin(45°)
Now, let's determine the torque exerted by the weight of the bicyclist about the center of the wheel. The torque formula is given by T = F * r * sin(θ), where:
- T is the torque
- F is the force
- r is the distance from the point of rotation to the line of action of the force
- θ is the angle between the force vector and the vector from the point of rotation
In this case, the torque due to the weight of the bicyclist can be calculated as follows:
Torque = (mg * sin(45°)) * 0.170 m
Since the chain must exert an equal and opposite torque to maintain equilibrium, we can set the torque due to the chain equal to the torque due to the weight of the bicyclist:
Torque due to chain = Torque due to weight of bicyclist
Let the force exerted by the chain be denoted as F_chain. We can rewrite the torque equation as follows:
(F_chain * 0.097 m) = (mg * sin(45°)) * 0.170 m
Now, solving this equation for F_chain, we get:
F_chain = (mg * sin(45°) * 0.170 m) / 0.097 m
Substituting the known values:
F_chain = (65 kg * 9.8 m/s^2 * sin(45°) * 0.170 m) / 0.097 m
Calculating this expression will give us the force that the chain must exert to keep the wheel from turning.