A cart loaded with bricks has a total mass of 25.2 kg and is pulled at constant speed by a rope. The rope is inclined at 21.6 ◦ above the horizontal and the cart moves 13.7 m on a horizontal floor. The coefficient of kinetic friction between ground and cart is 0.4 .
The acceleration of gravity is 9.8 m/s2 .
What is the normal force exerted on the cart by the floor? Answer in units of N.
How much work is done on the cart by the rope?
Answer in units of kJ.
The energy change due to friction is a loss of energy.
What is the energy change Wf due to fric- tion?
Answer in units of kJ.
(a)
Summing the components of the forces along the vertical direction provides
F• sinα + N - m•g = 0,
=> F• sinα = m•g - N.
For the horizontal direction we have
F•cosα – F(fr) = 0
F(fr) = k•N, => F•cosα = k•N,
F• sinα/ F•cosα = (m•g – N)/k•N,
tan α = (m•g – N)/k•N,
N = m•g/(k•tan +1) = 25.2•9.8/(0.4•tan21.6º +1) = 213 N.
(b)
If F•cosα = k•N,
F = k•N/cosα.
The work done on the cart is given by
W(fr) =F•s•cosα = k•N•s•cos α/cosα = =k•N•s = 0.4•213•13.7 = 1167 J =
=1.167 kJ.
(c)
The energy lost due to friction is just the work done by friction on the cart:
W(fr) = F(fr)•s•cos 180º = k•N•s•(-1) =
= - k•N•s = - 0.4•213•13.7 =
= - 1167 J = - 1.167 kJ.
To find the normal force exerted on the cart by the floor, you can use the equation:
Normal force (N) = mass (m) * gravitational acceleration (g) - weight of the cart (mg)
Given:
Mass of the cart (m) = 25.2 kg
Gravitational acceleration (g) = 9.8 m/s^2
Step 1: Calculate the weight of the cart (mg):
Weight of the cart = mass (m) * gravitational acceleration (g)
Weight of the cart = 25.2 kg * 9.8 m/s^2
Step 2: Plug the value of the weight of the cart into the equation for normal force:
Normal force (N) = mass (m) * gravitational acceleration (g) - weight of the cart (mg)
Normal force (N) = 25.2 kg * 9.8 m/s^2 - (25.2 kg * 9.8 m/s^2)
Simplify the equation to find the normal force exerted on the cart by the floor.
To find the work done on the cart by the rope, you can use the equation:
Work (W) = force (F) * distance (d) * cos(theta)
Given:
Force (F) = the force exerted by the rope
Distance (d) = 13.7 m (the distance the cart moves on a horizontal floor)
Cos(theta) = cosine of the angle between the force and the direction of motion
Step 1: Calculate the force exerted by the rope:
Force (F) = mass (m) * acceleration (a)
Given that the cart is moving at a constant speed, acceleration (a) is zero.
Step 2: Plug the values into the equation for work:
Work (W) = force (F) * distance (d) * cos(theta)
Work (W) = force (F) * 13.7 m * cos(21.6 degrees)