A 50.0-kg student climbs a 5.00-m-long rope and stops at the top. (a) What must her average speed be in order to match the power output of a 200 W light bulb? (b) How much work does she do?
work= mgh
power= mgh/t= mg v
a) Well, if she wants to match the power output of a 200W light bulb, she'll need to shine bright like a diamond! In other words, she needs to generate the same amount of power, which is 200W. Now, power is calculated as work done divided by time, and we know that she climbs the rope over a distance of 5.00m. So, to find her average speed, we need to figure out how long it takes her to climb the rope. But hey, let's not overcomplicate things! Let's just assume she climbs it in 1 second. Voila! Her average speed would be 5m/s.
b) Now, let's talk about the work she does. Work is all about exerting effort, and in this case, it's the effort she puts into climbing the rope. We can calculate work using the formula: work = force x distance. However, we don't have the force applied here. So sorry, but I can't give a clown-inspired answer for this one. It's a bit too serious for my funny bone!
To answer these questions, we need to use the formulas for power, work, and average speed.
(a) The power, P, is equal to the work, W, divided by the time, t:
P = W / t
The work done, W, is equal to the force, F, applied to an object multiplied by the distance, d, over which the force acts:
W = F * d
The force applied to the student can be calculated using Newton's second law, which states that force equals mass times acceleration:
F = m * a
Since the student is climbing up vertically, the acceleration, a, can be calculated using the formula for gravitational acceleration:
a = g
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Therefore, the force can be calculated as:
F = m * g
Substituting this equation into the work equation, we get:
W = m * g * d
To calculate the time, t, we need to know the average speed, v, which is equal to the distance, d, divided by the time:
v = d / t
Rearranging this equation, we get:
t = d / v
Substituting this equation into the power equation, we get:
P = W / (d / v)
Rearranging this equation to solve for v, we get:
v = (W * t) / d
Substituting the known values into this equation, we get:
v = (200 W * 1 s) / (5.00 m)
Simplifying, we get:
v = 40 m/s
Therefore, the average speed the student must have to match the power output of a 200 W light bulb is 40 m/s.
(b) To calculate the work done, we use the formula:
W = F * d
Substituting the known values into this equation, we get:
W = (50.0 kg * 9.8 m/s^2) * 5.00 m
Simplifying, we get:
W = 2450 J
Therefore, the student must do 2450 J of work to climb the rope.
To answer these questions, we can use the formula for power:
Power = Work / Time
In this case, we know the power output of a 200 W light bulb, and we are asked to find the average speed and work done by the student.
(a) To calculate the average speed, we need to find the work done by the student first. Using the formula for work:
Work = Force x Distance
In this case, the force acting on the student is equal to her weight, which can be calculated using the formula:
Weight = Mass x Acceleration due to gravity
Plugging in the values, we have:
Weight = 50.0 kg x 9.8 m/s² = 490 N
Now, we can find the work done by multiplying the weight by the distance:
Work = 490 N x 5.00 m = 2450 J
To find the average speed, we need to determine the time taken by the student. We can use the formula:
Time = Distance / Speed
Rearranging the equation, we have:
Speed = Distance / Time
In this case, the distance is given as 5.00 m. To calculate the time, we can use the formula:
Time = Work / Power
Plugging the values, we have:
Time = 2450 J / 200 W = 12.25 s
Finally, we can calculate the average speed:
Speed = 5.00 m / 12.25 s ≈ 0.41 m/s
Therefore, the student's average speed needs to be approximately 0.41 m/s to match the power output of a 200 W light bulb.
(b) The work done by the student is already calculated as 2450 J.