A cart full of packages weighing 180 pounds is rolling down a ramp 12 feet long at an incline of 25° (the force applied by gravity is equal to the weight of the package). Find the work done by gravity as the cart moves the length of the ramp. Assume that friction isn't a factor.
the vertical distance moved is
12 sin25° = 5.07 ft
Work = PE lost = 5.07 * 180 ft-lb
Thank you. I wasn't sure how to put the information given into equations.
To find the work done by gravity as the cart moves the length of the ramp, we need to calculate the force exerted by gravity on the cart and the displacement of the cart.
1. Calculate the force of gravity:
Since the force applied by gravity is equal to the weight of the cart, we can find the force of gravity using the formula:
Force of gravity = mass × gravity
Given that the cart weighs 180 pounds, we need to convert it to mass in order to use the formula. We know that 1 pound is approximately equal to 0.4536 kilograms.
So the mass of the cart in kilograms is:
Mass = 180 pounds × 0.4536 kg/pound
2. Calculate the force component acting along the incline:
The force of gravity can be divided into two components:
- The force component acting perpendicular to the incline, which doesn't contribute to the work done since it's orthogonal to the displacement.
- The force component acting along the incline, which is responsible for the work done by gravity.
We can calculate this component using the formula:
Force along incline = Force of gravity × sin(angle of incline)
3. Calculate the displacement of the cart:
The displacement of the cart is given as the length of the ramp, which is 12 feet. However, we need to convert it to meters in order to use it with the force calculations.
So the displacement of the cart in meters is:
Displacement = 12 feet × 0.3048 meters/foot
4. Calculate the work done by gravity:
The work done by gravity is given by the formula:
Work = Force along incline × displacement
Following these steps, you can calculate the work done by gravity.
To find the work done by gravity as the cart moves the length of the ramp, we need to calculate the force of gravity acting on the cart and then multiply it by the distance the cart moves.
First, let's find the force of gravity acting on the cart. The force of gravity is equal to the weight of the cart, which is given as 180 pounds. Since the force applied by gravity is equal to the weight of an object, we can assume that the force of gravity on the cart is 180 pounds.
Next, we need to find the component of the gravitational force that acts in the direction of the ramp. This component is equal to the force of gravity multiplied by the sine of the angle of incline (25°). So, the component of the gravitational force in the direction of the ramp is:
Force of gravity * sin(angle of incline)
The angle of incline is given as 25°, so we can plug the values into the formula:
180 pounds * sin(25°)
Now, we can calculate the work done by gravity. Work is equal to the force applied multiplied by the distance traveled. In this case, the force applied by gravity is the component of the gravitational force in the direction of the ramp, and the distance traveled is the length of the ramp, which is 12 feet.
So, the work done by gravity is:
180 pounds * sin(25°) * 12 feet
Using a calculator, we can find the decimal value of sin(25°), which is approximately 0.4226. Plugging this value into the equation:
180 pounds * 0.4226 * 12 feet
Now, we can calculate the final answer:
work = 180 pounds * 0.4226 * 12 feet
≈ 903.504 foot-pounds
Therefore, the work done by gravity as the cart moves the length of the ramp is approximately 903.504 foot-pounds.