2. You and your bicycle have combined mass 80.0 kg. When you reach the base of a bridge, you are traveling along !he road at 5.00 mls (Fig. 6.35). At the top of the bridge, you have climbed a vertical distance of 5.20 m and have slowed to 1.50 m/s. You can ignore work done by friction and any inefficiency in !he bike or your legs. (a) What is !he total work done on you and your bicycle when you go from the base to the top of !he bridge? (b) How much work have you done with the force you apply to !he pedals?
To find the total work done on you and your bicycle when going from the base to the top of the bridge, we need to consider the change in kinetic energy and the change in potential energy.
(a) The work done on an object is equal to the change in its total mechanical energy. This can be found using the equation:
Work = ΔKE + ΔPE
First, let's calculate the change in kinetic energy (ΔKE).
The initial kinetic energy (KEi) can be found using the equation:
KEi = (1/2) * mass * velocity^2
Plugging in the values:
KEi = (1/2) * 80.0 kg * (5.00 m/s)^2
KEi = 1000 J
The final kinetic energy (KEf) can be found using the equation:
KEf = (1/2) * mass * velocity^2
Plugging in the values:
KEf = (1/2) * 80.0 kg * (1.50 m/s)^2
KEf = 90 J
The change in kinetic energy is given by:
ΔKE = KEf - KEi
ΔKE = 90 J - 1000 J
ΔKE = -910 J
Next, let's calculate the change in potential energy (ΔPE).
The initial potential energy (PEi) is zero since it is measured at the base of the bridge.
The final potential energy (PEf) can be found using the equation:
PEf = mass * gravity * height
Plugging in the values:
PEf = 80.0 kg * 9.8 m/s^2 * 5.20 m
PEf = 4112 J
The change in potential energy is given by:
ΔPE = PEf - PEi
ΔPE = 4112 J - 0 J
ΔPE = 4112 J
Now, the total work done on you and your bicycle is:
Work = ΔKE + ΔPE
Work = -910 J + 4112 J
Work = 3202 J
Therefore, the total work done on you and your bicycle when going from the base to the top of the bridge is 3202 Joules.
(b) The work done with the force applied to the pedals is the work done to change the kinetic energy. In this case, the work done by the force applied to the pedals is the negative value of the change in kinetic energy.
Work (pedals) = -ΔKE
Work (pedals) = -(-910 J)
Work (pedals) = 910 J
Therefore, the work done with the force you apply to the pedals is 910 Joules.