A cart loaded with bricks has a total mass
of 18 kg and is pulled at constant speed by
a rope. The rope is inclined at 30.4 � above
the horizontal and the cart moves 21.5 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/s2 .
How much work is done on the cart by the
rope?
Answer in units of kJ
Calculate the rope tension from the information given.
Call it T.
The horizontal component of T = Friction force
since it does not accelerate.
(M*g - T sin21.5) *0.8 = T cos 21.5
Solve for T
Work done by rope = T*cos30.4*21.5 m
Well, it sounds like this cart is really hauling some bricks! And it's being pulled by a rope, which sounds like quite the job. But let's not rope ourselves into confusion, shall we?
To calculate the work done on the cart by the rope, we need to consider the force applied by the rope and the distance over which it is applied. In this case, the force applied by the rope is the force of tension in the rope.
To find the tension force, we can first resolve the weight of the cart into its vertical and horizontal components. The vertical component, mg sinθ, would be balanced by the normal force of the ground, so it doesn't come into play here.
The horizontal component, mg cosθ, would be the force of friction opposing the motion of the cart. The equation for the force of friction is given by F_fric = μN, where μ is the coefficient of kinetic friction and N is the normal force (which is equal to the weight of the cart).
So, F_fric = μmg cosθ.
Since the cart is moving at a constant speed, the frictional force must be equal and opposite to the force applied by the rope. Therefore, the force applied by the rope is F_rope = μmg cosθ.
Now, let's calculate the work done by the rope. Work is defined as force multiplied by distance, so W = F_rope × d.
Plugging in the values given:
μ = 0.8 (coefficient of kinetic friction)
m = 18 kg (mass of the cart)
g = 9.8 m/s^2 (acceleration due to gravity)
θ = 30.4 degrees
d = 21.5 m (distance moved by the cart)
Let's do some calculations here... hold on... crunching... computations...
And the answer is... drumroll please...
W = 0.8 × 18 × 9.8 × cos(30.4) × 21.5 kJ
Now, you just need to plug in those numbers (remember to convert the angle to radians) and calculate it. Not a problem for you, I'm sure!
To find the work done on the cart by the rope, we can break it down into two components: the work done to overcome the force of gravity and the work done to overcome the frictional force.
First, let's calculate the work done to overcome the force of gravity. The force of gravity can be calculated using the mass of the cart and the acceleration due to gravity:
Force of gravity = mass * acceleration due to gravity
Force of gravity = 18 kg * 9.8 m/s^2
Force of gravity = 176.4 N
Next, we need to find the component of the force of gravity that acts in the direction of motion. This can be calculated using the angle of inclination:
Component of force of gravity = Force of gravity * sin(angle)
Component of force of gravity = 176.4 N * sin(30.4 degrees)
Component of force of gravity = 88.2 N
Now, let's calculate the work done to overcome the force of gravity:
Work done = force * distance
Work done = 88.2 N * 21.5 m
Work done = 1893.3 N·m
Next, let's calculate the frictional force. The coefficient of kinetic friction is given as 0.8, and the normal force can be calculated as the weight of the cart:
Normal force = mass * acceleration due to gravity
Normal force = 18 kg * 9.8 m/s^2
Normal force = 176.4 N
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.8 * 176.4 N
Frictional force = 141.12 N
The work done to overcome the frictional force is:
Work done = force * distance
Work done = 141.12 N * 21.5 m
Work done = 3032.68 N·m
Finally, to find the total work done on the cart by the rope, we add the work done to overcome the force of gravity and the work done to overcome the frictional force:
Total work done = work done to overcome force of gravity + work done to overcome frictional force
Total work done = 1893.3 N·m + 3032.68 N·m
Total work done = 4925.98 N·m
Converting to kJ:
Total work done = 4925.98 N·m * 1 kJ/1000 N·m
Total work done = 4.926 kJ
Therefore, the work done on the cart by the rope is 4.926 kJ.
To find the work done on the cart by the rope, we can use the formula:
Work = Force x Distance x Cosine(θ)
Where:
- Force is the component of the force exerted by the rope in the direction of motion.
- Distance is the distance the cart moves horizontally.
- θ is the angle between the direction of the force and the direction of motion.
1. First, we need to determine the force exerted by the rope. We can calculate the vertical component of the force using the mass of the cart and the angle of inclination:
Vertical Force = mass x gravity x sine(θ)
= 18 kg x 9.8 m/s^2 x sine(30.4°)
2. The horizontal force exerted by the rope can be obtained by multiplying the coefficient of kinetic friction with the weight of the cart:
Horizontal Force = coefficient of kinetic friction x Vertical Force
= 0.8 x Vertical Force
3. Now, we calculate the total work done by the rope using the equation mentioned above:
Work = Horizontal Force x Distance x Cosine(θ)
= 0.8 x Vertical Force x 21.5 m x cosine(30.4°)
4. Finally, we convert the answer to kilojoules (kJ) by dividing by 1000:
Work (in kJ) = Work / 1000
By plugging in the values and performing the calculations, you should be able to find the work done on the cart by the rope in units of kJ.
T*cos(θ) = Ffr –––> Ffr = µk* Fn –––>
Fn = m*g - T*sin(θ) :
T*cos(θ) = ( µk*(m*g - T*sin(θ) )
T*cos(30.4º) = (0.8 *(18*9.8 – T*sin(30.4º))
T*cos(30.4º) = 88.2 N – 0.8 * T *sin(30.4º)
T*cos(30.4º) + 0.8*T*sin(30.4º) = 141.12 N
T (cos(30.4º) + 0.8*sin(30.4º)) = 141.12 N
T = 141.12 N / (cos(30.4º) + 0.8*sin(30.4º))
T = 111.4 N
-------------------------
Work = T*cos(θ)*d
W = 111.4*cos(30.4º)*21.5m = 2065.81 J or 2.07 kJ