A 45 kg cart is pushed up a ramp a length of 5.8 m from rest, attaining a speed of 2.6 m/s
at the top of the ramp, which is 1.7 m high. The coefficient of friction between the cart
and the ramp is 0.13.
a) Determine the work done against:
1) gravity 2) inertia 3) friction.
b) What force was used to push the cart?
c) What power was used to move the cart?
To determine the work done against gravity, inertia, and friction, as well as the force used to push the cart and the power used to move the cart, we'll need to use the following formulas:
1) Work done against gravity: W_gravity = m * g * h
2) Work done against inertia: W_inertia = (1/2) * m * v^2
3) Work done against friction: W_friction = μ * m * g * d
4) Force used to push the cart: F_push = m * a
5) Power used to move the cart: P = W / t
Where:
m = mass of the cart (45 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the ramp (1.7 m)
v = speed of the cart (2.6 m/s)
μ = coefficient of friction (0.13)
d = length of the ramp (5.8 m)
a = acceleration of the cart
W = work done
t = time
Let's calculate each part step by step:
a) Determine the work done against:
1) gravity: W_gravity = m * g * h
W_gravity = 45 kg * 9.8 m/s^2 * 1.7 m
W_gravity ≈ 713.19 J (Joules)
2) inertia: W_inertia = (1/2) * m * v^2
W_inertia = (1/2) * 45 kg * (2.6 m/s)^2
W_inertia ≈ 170.1 J (Joules)
3) friction: W_friction = μ * m * g * d
W_friction = 0.13 * 45 kg * 9.8 m/s^2 * 5.8 m
W_friction ≈ 344.07 J (Joules)
b) What force was used to push the cart?
To determine the force used to push the cart, we need to find the acceleration of the cart.
We can use the equation v^2 = u^2 + 2ad, where u is the initial velocity (0 m/s) and a is the acceleration.
v^2 = u^2 + 2ad
(2.6 m/s)^2 = (0 m/s)^2 + 2 * a * 5.8 m
6.76 m^2/s^2 = 11.6 m * a
a ≈ 0.58 m/s^2
F_push = m * a
F_push = 45 kg * 0.58 m/s^2
F_push ≈ 26.1 N (Newtons)
c) What power was used to move the cart?
Power is the rate at which work is done, so we need to know the time it took to move up the ramp.
Unfortunately, the time is not provided in the question. If you have the time, please provide it so we can calculate the power using P = W / t.