How much work is done by the orderly pushing a 85 kg person up a 5 m ramp inclined at an angle of 15° to the horizontal?
I did W = (85)(9.8)(5)cos15, but to get the correct answer you are supposed to use
(85)(9.8)(5)sin15 instead.
Why do we have to use sin?
5 sin(15º) gives the vertical displacement (work against gravity)
cosine gives the horizontal displacement
He is doing work against gravity, vertical. The ramp is at75 deg to the vertical.
work=85(9.8)5cos75 = if you wish...85*9.8*5sin15
In this scenario, the correct approach is to use the sine function because the angle of 15° is the angle of inclination of the ramp, which is the angle between the ramp and the horizontal plane.
The work done is calculated using the equation W = Fd cosθ, where F represents the force applied, d represents the displacement, and θ represents the angle between the force and the displacement.
In this case, the force applied by the orderly is directed parallel to the inclined ramp, which is perpendicular to the vertical component of the person's weight. Therefore, we need to calculate the component of the person's weight that acts parallel to the ramp, which can be found using the sine function.
Using the sine function, the vertical component of the person's weight is given by (85 kg)(9.8 m/s^2)sin(15°). This component represents the force applied by gravity along the ramp, which is equal in magnitude and opposite in direction to the force exerted by the orderly.
So, in the equation W = Fd cosθ, the force F should be replaced with (85 kg)(9.8 m/s^2)sin(15°), resulting in the correct calculation of the work done by the orderly.
To understand why we use the sin function to calculate the work done in this scenario, let's first clarify the concept of work and how it relates to forces and displacement.
Work is defined as the transfer of energy that occurs when a force is applied to move an object over a distance. Mathematically, the equation for work is:
Work = Force x Distance x cos(angle)
In this case, the force being exerted is the force applied by the orderly to push the person up the ramp.
Now, let's consider the inclined ramp. When an object is on an inclined plane, its weight can be resolved into two components: one perpendicular to the ramp (normal force) and one parallel to the ramp (force due to gravity along the ramp surface).
In this scenario, the force parallel to the ramp (force due to gravity) is the force the orderly needs to overcome to push the person up the ramp. This force along the ramp is given by:
Force along ramp = Weight x sin(angle)
Here, the weight is the gravitational force acting on the person, which is equal to their mass multiplied by the acceleration due to gravity (9.8 m/s^2).
Therefore, the correct equation for the force along the ramp is:
Force along ramp = (mass x gravity) x sin(angle)
Now, to calculate the work done by the orderly, we need to substitute the force along the ramp and the distance into the work equation:
Work = (Force along ramp) x Distance x cos(angle)
Since we already determined that the force along the ramp is (mass x gravity) x sin(angle), the correct equation becomes:
Work = (mass x gravity x sin(angle)) x Distance x cos(angle)
Notice that in this equation, the sine of the angle (sin(angle)) is multiplied by the mass, while the cosine of the angle (cos(angle)) is multiplied by the distance. The reason for this is that the force along the ramp (parallel to the ramp) depends on the sine of the angle, whereas the displacement (distance) along the ramp depends on the cosine of the angle.
Hence, to calculate the work done by the orderly pushing the person up the 5 m ramp inclined at an angle of 15°, we should use:
Work = (mass x gravity x sin(angle)) x Distance x cos(angle)
Substituting the given values of mass (85 kg), gravity (9.8 m/s^2), angle (15°), and distance (5 m) into this equation will give you the correct answer.