please help with cart thing
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?
thanks for your help :)
To solve this problem, we can break it down into several steps:
Step 1: Determine the work done against gravity.
The work done against gravity can be calculated using the formula: work = force * distance * cos(theta).
In this case, the force is the weight of the cart, which can be calculated using the formula: weight = mass * gravitational acceleration.
First, let's find the force of gravity acting on the cart:
mass = 45 kg
gravitational acceleration = 9.8 m/s^2
weight = mass * gravitational acceleration
Next, let's calculate the distance and the angle (theta) at which the cart is moved.
distance = 5.8 m
theta = the angle between the ramp and the horizontal (in this case, it's the angle of the ramp which makes an inclined plane).
Now we can calculate the work done against gravity using the formula mentioned earlier.
Step 2: Determine the work done against inertia.
The work done against inertia can be calculated using the formula: work = 0.5 * mass * velocity^2.
In this case, the mass is 45 kg, and the velocity can be calculated using the following equation based on conservation of energy:
Potential Energy at the top = Kinetic Energy at the bottom
First, let's find the potential energy at the top:
potential energy = mass * gravitational acceleration * height
Next, let's find the kinetic energy at the bottom:
kinetic energy = 0.5 * mass * velocity^2
Now, we can equate the potential energy and kinetic energy to solve for velocity.
Once we have the velocity, we can calculate the work done against inertia using the formula mentioned earlier.
Step 3: Determine the work done against friction.
The work done against friction can be calculated using the formula: work = force of friction * distance.
In this case, the force of friction can be calculated using the formula: force of friction = coefficient of friction * weight.
Once we have the force of friction, we can calculate the work done against friction using the formula mentioned earlier.
Step 4: Determine the force used to push the cart.
The force used to push the cart can be calculated by summing up the forces acting on the cart, which in this case are the force of friction and the force of gravity.
Step 5: Determine the power used to move the cart.
Power is defined as the rate at which work is done. It can be calculated using the formula: power = work / time.
Since we do not have the time in this problem, it's not possible to directly calculate power. However, we can use the work done against gravity or the work done against friction and substitute it into the power formula to find the power used to move the cart.
s = slope angle
sin s = 1.7/5.8
so s = 17 deg
so cos s = .956
1) m g h = 45 * 9.81 * 1.7
2) (1/2) m v^2 = (1/2) (45)(2.6)^2
3) normal force = m g cos T = 45*9.81*.956 = 422 N
friction force = .13 * 422 = 54.9 N
work against friction = 54.9*5.8
b) Total work done = increase in potential energy (1) + increase in kinetic energy (2) + work against friction (3)
so
F = that total work done / 5.8
c) power = total work done/ time up ramp
get time from average speed which is 2./2 = 1.3 m/s
t = 5.8 / 1.3
so
power = that total work done above /t