A sled loaded with bricks has a total mass of 23.0 kg and is pulled at constant speed by a rope inclined at 40.0o above the horizontal. The sledge moves a distance of 12.0 m on a horizontal surface. The coefficient of kinetic friction between the sledge and surface is 0.350 .
(a) What is the tension in the rope?
(b) How much work is done by the rope on the sled?
(c) What is the magnitude of the force of friction?
To find the tension in the rope, we can first determine the force of gravity acting on the sled:
Force of gravity (Fg) = mass (m) x acceleration due to gravity (g)
Fg = 23.0 kg x 9.8 m/s^2 = 225.4 N
The vertical component of the force of gravity can be found by multiplying the force of gravity by the sine of the angle of incline:
Vertical component of Fg = Fg x sin(40.0o)
Vertical component of Fg = 225.4 N x sin(40.0o) = 144.9 N
Since the sled is moving at constant speed, the tension in the rope must be equal to the horizontal component of the force of gravity. This can be found using cosine:
Tension = Fg x cos(40.0o)
Tension = 225.4 N x cos(40.0o) = 172.3 N
(a) The tension in the rope is 172.3 N.
To find the work done by the rope on the sled, we can use the equation:
Work = force x distance x cosine(angle)
Since the sled is moving horizontally, the angle between the force and the displacement is 0o, so the cosine of 0o is 1.
Work = Tension x distance
Work = 172.3 N x 12.0 m = 2068.7 J
(b) The work done by the rope on the sled is 2068.7 J.
To find the magnitude of the force of friction, we can use the equation:
Force of friction = coefficient of friction x normal force
The normal force is equal to the vertical component of the force of gravity:
Normal force = Vertical component of Fg = 144.9 N
Force of friction = 0.350 x 144.9 N = 50.7 N
(c) The magnitude of the force of friction is 50.7 N.
To find the answers to these questions, we can use the concepts of Newton's laws of motion and work-energy theorem. Let's break down each part and see how we can get the answer.
(a) What is the tension in the rope?
To find the tension in the rope, we need to consider the forces acting on the sled. The tension in the rope will be equal in magnitude to the force required to overcome the force of friction and the force component due to gravity along the inclined plane.
First, let's calculate the force of gravity acting on the sled. We can do this by finding the component of the gravitational force that acts parallel to the inclined plane:
Force_parallel = m * g * sin(theta)
where m is the mass of the sled (23.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and theta is the angle of the incline (40.0 degrees).
Next, we need to find the force of friction. The force of friction can be calculated using the coefficient of kinetic friction (0.350) and the normal force, which is equal to the weight of the sled:
Force_friction = coefficient_of_friction * force_normal
where force_normal = m * g.
Finally, to find the tension in the rope, we need to sum up the forces along the horizontal direction:
Tension = Force_parallel + Force_friction
Substitute the values into the equations and calculate to get the answer.
(b) How much work is done by the rope on the sled?
The work done by the rope on the sled can be found using the formula:
Work = force_parallel * displacement
where displacement is the distance the sled moves on the horizontal surface (12.0 m).
Substitute the calculated value for force_parallel and the given value for displacement into the equation to find the work done by the rope.
(c) What is the magnitude of the force of friction?
The magnitude of the force of friction can be found using the formula:
Force_friction = coefficient_of_friction * force_normal
where force_normal = m * g.
Substitute the values into the equation to calculate the magnitude of the force of friction.