A cyclist is attached to a rope which is looped over a pulley (disc of mass 10kg and radius 0.5m). The rope is then
attached to a hanging block of mass 5kg. What force must the cyclist generate such that the block will move with an
acceleration of 2.25 m/s2. Let’s make the assumption that only one wheel of the bicycle is necessary to generate the
force (we can ignore the front wheel) and that all the force will manifest itself in rolling friction. The mass of the
bicycle and the cyclist is 90kg. The surface has a coefficient of static friction of 0.65 and kinetic friction of 0.25.
To determine the force the cyclist must generate in order to move the block with a specific acceleration, we need to consider the forces acting on the system.
1. Start by calculating the gravitational force acting on the hanging block:
F_gravity = mass_block * g
where mass_block = 5kg and g = 9.8 m/s^2 (acceleration due to gravity)
2. Since the block is accelerating, there must be a net force acting on it. Let's call this force F_net.
3. The force required to overcome rolling friction can be calculated using the formula:
F_friction = coefficient_friction * normal_force
where coefficient_friction is the coefficient of kinetic friction between the wheel and the surface, and normal_force is the force exerted by the block on the wheel.
4. The normal force can be calculated as the weight of the pulley plus the force exerted by the cyclist:
normal_force = F_gravity + (mass_pulley * g)
5. The force exerted by the cyclist, F_cyclist, can be calculated as the difference between the net force and the frictional force:
F_cyclist = F_net - F_friction
6. The net force can be calculated using Newton's second law:
F_net = mass_block * acceleration
where acceleration = 2.25 m/s^2 (given in the question)
7. Now let's calculate the frictional force using the coefficient of kinetic friction:
F_friction = coefficient_kinetic_friction * normal_force
where coefficient_kinetic_friction = 0.25 (given in the question)
8. Finally, substitute the values into the equations to find the force the cyclist must generate:
F_cyclist = (mass_block * acceleration) - (coefficient_kinetic_friction * (F_gravity + (mass_pulley * g)))
Substitute the given values and solve the equation to find the force the cyclist must generate.