A 200-kg cart is pushed slowly up an incline. How much work does the pushing force do in moving the cart up to a platform 1.5 m above the starting point if friction is negligible?
(Use above problem) If the distance along the incline to the platform is 7 m and a friction force of 150 N opposes the motion.
The work done by the pushing force is equal to the change in the cart's potential energy. The potential energy of the cart is equal to its mass multiplied by the acceleration due to gravity and the height of the platform.
Work = (200 kg)(9.8 m/s2)(1.5 m) = 2940 J
The work done against friction is equal to the force of friction multiplied by the distance along the incline.
Work against friction = (150 N)(7 m) = 1050 J
Therefore, the total work done by the pushing force is equal to 2940 J - 1050 J = 1890 J.
To determine the work done by the pushing force in moving the cart up the incline, we can use the equation:
Work = Force × Displacement × cos(θ)
Where:
- Work is the amount of work done (in joules, J)
- Force is the force applied (in newtons, N)
- Displacement is the distance traveled along the incline (in meters, m)
- θ (theta) is the angle between the applied force and the direction of displacement
In the first scenario where friction is negligible, the pushing force is the only force acting on the cart, and the angle between the force and the displacement is 0 degrees (since they are parallel). Therefore, θ = 0.
Given:
- Mass of the cart (m) = 200 kg
- Displacement (along the incline) (d) = 7 m
- Height (h) of the platform above the starting point = 1.5 m
1. Calculate the work done against gravity:
The force due to gravity can be found using the formula: Force_gravity = mass × acceleration_due_to_gravity.
Acceleration_due_to_gravity = 9.8 m/s² (approximate value on Earth)
Force_gravity = 200 kg × 9.8 m/s² = 1960 N
The work done against gravity can be calculated as:
Work_gravity = Force_gravity × Displacement × cos(θ)
Since θ = 0, the cosine of 0 is 1.
Work_gravity = 1960 N × 7 m × 1 = 13,720 J
2. Calculate the work done by the friction force:
Given:
- Friction force (F_friction) = 150 N
- Displacement (d) = 7 m
Work_friction = F_friction × Displacement × cos(θ)
Again, since θ = 0, the cosine of 0 is 1.
Work_friction = 150 N × 7 m × 1 = 1050 J
3. Calculate the work done by the pushing force:
Since friction is negligible, the work done by the pushing force is equal to the net work done on the cart, which accounts for both gravitational work and work against friction.
Work_pushing = Work_gravity + Work_friction
Work_pushing = 13,720 J + 1050 J = 14,770 J
Therefore, the pushing force does 14,770 joules of work in moving the cart up to a platform 1.5 m above the starting point when friction is negligible.
To find the work done by the pushing force in moving the cart up to the platform, we need to calculate the net force and the distance moved.
Given:
Mass of the cart (m) = 200 kg
Height of the platform (h) = 1.5 m
Since friction is negligible, the only force opposing the motion is the gravitational force acting down the incline. The work done by the pushing force will overcome this force.
First, let's calculate the gravitational force acting down the incline:
Gravitational force (Fg) = mass (m) x acceleration due to gravity (g)
The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Fg = 200 kg x 9.8 m/s^2
Fg = 1960 N
Since friction is negligible, the net force acting on the cart is the component of the gravitational force parallel to the incline.
Net force (Fnet) = Fg * sin(theta)
To find the angle theta, we can use the relationship between the height (h) and the distance along the incline (d):
sin(theta) = h / d
Given:
Distance along the incline (d) = 7 m
sin(theta) = 1.5 m / 7 m
sin(theta) ≈ 0.2143
Now we can calculate the net force:
Fnet = 1960 N * 0.2143
Fnet ≈ 420.28 N
Finally, we can calculate the work done by the pushing force using the formula:
Work (W) = Fnet * distance (d)
W = 420.28 N * 7 m
W ≈ 2941.96 J
Therefore, the work done by the pushing force in moving the cart up to the platform is approximately 2941.96 Joules.