A cart loaded with bricks has a total mass of 22.3 kg and is pulled at constant speed by a rope. The rope is inclined at 27.3◦degrees above the horizontal and the cart moves 9.3 m on a horizontal surface. The coefficient of kinetic friction between ground and cart is 0.442. The acceleration of gravity is 9.8 m/s^2. How much work is done on the cart by the rope?
Answer in units of J.
Fc = m*g = 22.3kg * 9.8N/kg = 218.54 N = Wt. of cart.
Fk = u*Fc = 0.442 * 218.54 = 96.6 N. =
Force of kinetic friction.
Fap*cos27.3-Fk = m*a
Fap*cos27.3-96.6 = m*0 = 0
Fap*cos27.3 = 96.6 N.
W=Fap*cos27.3 * d = 96.6 * 9.3=898.3 J.
Fap = Force applied = Tension in rope.
Correction:
Fk = u*Fc-u*Fap*sin27.3
Fk = 0.442*218.54 - 0.442*Fap*sin27.3
Fk = 96.6 - 0.203Fap
Fap*cos27.3-(96.6-0.203Fap) = m*a
0.889Fap - 96.6 + 0.203Fap = m*0 1.092Fap = 96.6
Fap = 88.5 N. = Force applied.
W = Fap*cos27.3 * d
W = 88.5*cos27.3 * 9.3 = 731.3 J.
To find the work done on the cart by the rope, we need to calculate the force exerted by the rope and then multiply it by the distance over which the force is applied.
First, let's find the force exerted by the rope. The force exerted by the rope can be divided into two components: a horizontal component and a vertical component.
The vertical component of the force is equal to the weight of the cart, which can be calculated as follows:
Weight = mass * gravity
Weight = 22.3 kg * 9.8 m/s^2
Weight = 218.54 N (Newtons)
The horizontal component of the force is equal to the force required to overcome the kinetic friction between the cart and the ground. This can be calculated as follows:
Frictional force = coefficient of kinetic friction * normal force
Normal force = mass * gravity * cos(theta)
Frictional force = 0.442 * (22.3 kg * 9.8 m/s^2 * cos(27.3 degrees))
Frictional force = 82.51 N
Now, let's find the force exerted by the rope:
Force by rope = sqrt(horizontal force^2 + vertical force^2)
Force by rope = sqrt((Frictional force)^2 + (Weight)^2)
Force by rope = sqrt((82.51 N)^2 + (218.54 N)^2)
Force by rope = 237.17 N
Finally, let's calculate the work done by the rope:
Work = force * distance
Work = 237.17 N * 9.3 m
Work = 2205.68 J
Therefore, the work done on the cart by the rope is 2205.68 Joules (J).