A cart loaded with bricks has a total mass
of 27.7 kg and is pulled at constant speed by
a rope. The rope is inclined at 23.7
◦
above
the horizontal and the cart moves 23.1 m on
a horizontal floor. The coefficient of kinetic
friction between ground and cart is 0.8 .
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 kJ
12
To find the work done on the cart by the rope, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy.
First, let's calculate the force exerted by the rope on the cart. The gravitational force pulling the cart downwards is given by the equation:
F_gravity = m * g
where m is the mass of the cart and g is the acceleration due to gravity.
So, substituting the values given:
m = 27.7 kg
g = 9.8 m/s^2
F_gravity = 27.7 kg * 9.8 m/s^2
F_gravity = 271.46 N
Since the cart is being pulled at a constant speed, the force applied by the rope must balance the force of friction. The frictional force can be calculated using the equation:
F_friction = μ * F_normal
where μ is the coefficient of kinetic friction and F_normal is the normal force. The normal force is equal to the gravitational force acting perpendicular to the incline:
F_normal = F_gravity * cosθ
where θ is the angle of inclination.
Substituting the values given:
μ = 0.8
θ = 23.7 degrees (convert to radians)
First, we need to convert the angle from degrees to radians:
θ = 23.7 * π/180
θ = 0.4137 radians
Then we can calculate the normal force:
F_normal = 271.46 N * cos(0.4137 radians)
F_normal = 217.182 N
Now, let's calculate the frictional force:
F_friction = 0.8 * 217.182 N
F_friction = 173.746 N
Since the cart is moving at constant speed, the applied force by the rope must balance the frictional force. Therefore, the force applied by the rope is equal to the frictional force:
F_rope = F_friction
F_rope = 173.746 N
Finally, we can calculate the work done on the cart by the rope using the equation:
Work = force * distance * cosθ
However, since the cart moves on a horizontal floor, the angle θ is 0 degrees. Therefore, the cosine of 0 degrees is equal to 1.
Work = F_rope * distance
Work = 173.746 N * 23.1 m
Work = 4011.0446 N·m
To convert the work to kilojoules (kJ), divide by 1000:
Work = 4011.0446 N·m / 1000
Work = 4.011 kJ
Therefore, the work done on the cart by the rope is 4.011 kJ.