In mid-nineteenth century England horses were often used to pull barges along canals. The horse, naturally, trudged along a path parallel to the canal and so the rope to the barge was at an angle to the motion of the barge.
If such a horse was pulling with a force of 175 N at a speed of 12.80 Km/hr and the angle of the rope with respect to the canal direction was 25.0 degrees how much work (in KJ) does the horse do in 9.1 minutes?
To calculate the work done by the horse, we need to use the formula:
Work = Force * Distance * Cosine(theta)
First, let's convert the speed from kilometers per hour to meters per second, as it is a more appropriate unit for calculations involving force and distance.
Given:
Force (F) = 175 N
Speed (v) = 12.80 km/hr
Time (t) = 9.1 minutes
Speed (v) in meters per second:
v = (12.80 km/hr) * (1000 m/km) * (1 hr/3600 s)
v ≈ 3.56 m/s
Now, let's calculate the distance covered by the horse in 9.1 minutes. We need to convert the time into seconds, as the speed is given in meters per second.
Time (t) in seconds:
t = 9.1 minutes * 60 seconds/minute
t = 546 seconds
Distance (d) covered by the horse:
d = v * t
d = 3.56 m/s * 546 s
d ≈ 1941.76 meters
Next, we need to calculate the angle between the direction of the force and the direction of motion of the barge, which is given as 25.0 degrees.
Theta (θ) in radians:
θ = 25.0 degrees * (π radians/180 degrees)
θ ≈ 0.4363 radians
Now, we can calculate the work done by the horse using the formula mentioned above.
Work (W) = F * d * Cos(θ)
W = 175 N * 1941.76 meters * Cos(0.4363 radians)
Cosine of 0.4363 radians is approximately 0.9045.
W ≈ 175 N * 1941.76 meters * 0.9045
W ≈ 313,309.44 N·m
Finally, let's convert the work to kilojoules (KJ) by dividing by 1000.
Work (W) in KJ:
W = 313,309.44 N·m / 1000
W ≈ 313.31 KJ
Therefore, the horse does approximately 313.31 kilojoules of work in 9.1 minutes.
To find the work done by the horse, we can use the formula:
Work = Force * Distance * cos(theta)
In this case, the force is given as 175 N, the distance is the speed multiplied by time, and theta is the angle of the rope with respect to the canal direction.
First, let's convert the speed from Km/hr to m/s.
12.80 Km/hr * (1000 m / 1 Km) * (1 hr / 3600 s) = 3.56 m/s
Now, let's convert the time from minutes to seconds.
9.1 minutes * (60 s / 1 min) = 546 s
Next, let's calculate the distance traveled by the horse.
Distance = Speed * Time = 3.56 m/s * 546 s = 1940.76 m
Now, let's calculate the work done.
Work = 175 N * 1940.76 m * cos(25.0 degrees)
To calculate the cosine of 25.0 degrees, we need to convert it to radians.
25.0 degrees * (pi / 180 degrees) = 0.4363 radians
Now, let's substitute the values into the formula.
Work = 175 N * 1940.76 m * cos(0.4363 radians)
Work ≈ 165,288 J
Finally, let's convert the work from Joules to Kilojoules.
Work = 165,288 J * (1 KJ / 1000 J)
Work ≈ 165.29 KJ
Therefore, the horse does approximately 165.29 KJ of work in 9.1 minutes.