A mover has to move a heavy sofa of mass 49.0 kg to the second floor of the house. As he pulls on the rope tied to the sofa, he makes sure that the rope is parallel to the surface of the ramp, which is at 30.0° to the horizontal. If the coefficient of kinetic friction between the sofa and the ramp is 0.320, and the sofa has an acceleration of 0.800 m/s2, find the tension in the rope.
See previous post: Thu, 9-24-15, 10:35 PM.
Well, it seems like this mover has got himself into a bit of a sofa struggle! Let's see if we can help him out.
First, let's break down the forces acting on the sofa. We have the gravitational force pulling it downward, which can be calculated as F_grav = mass * gravity. So, F_grav = 49.0 kg * 9.8 m/s^2 = 480.2 N (Newton is a unit of force, by the way).
Now let's find the normal force, which is the force exerted by the ramp perpendicular to its surface. Since the ramp is at an angle of 30.0° to the horizontal, the normal force can be found as F_normal = F_grav * cos(30.0°). Plugging in the numbers, F_normal = 480.2 N * cos(30.0°) = 416.6 N.
Next, we need to find the force of friction acting on the sofa. This force can be calculated as F_friction = coefficient of kinetic friction * F_normal. So, F_friction = 0.320 * 416.6 N = 133.3 N.
Now, let's find the tension in the rope. The tension force is equal to the sum of the force of friction and the force required to accelerate the sofa. So, tension = F_friction + mass * acceleration. Plugging in the values, tension = 133.3 N + (49.0 kg * 0.800 m/s^2) = 171.3 N.
So, the tension in the rope is a whopping 171.3 Newtons! That's one strong rope the mover's got there. Keep up the good work, Mr. Mover!
To find the tension in the rope, we need to consider the forces acting on the sofa.
1. Gravity force (Fg): This is the force pulling the sofa downward and can be calculated using the formula Fg = m * g, where m is the mass of the sofa (49.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = (49.0 kg) * (9.8 m/s^2) = 480.2 N
2. Normal force (Fn): This is the force exerted by the ramp perpendicular to the surface and can be calculated using the formula Fn = m * g * cos(theta), where theta is the angle of the ramp (30.0°).
Fn = (49.0 kg) * (9.8 m/s^2) * cos(30.0°) = 423.7 N
3. Friction force (Ff): This is the force opposing the motion and can be calculated using the formula Ff = coefficient of friction * Fn. The coefficient of kinetic friction is given as 0.320.
Ff = (0.320) * (423.7 N) = 135.5 N
4. Net force (Fnet): This is the force responsible for the acceleration and can be calculated using the formula Fnet = m * a, where a is the acceleration of the sofa (0.800 m/s^2).
Fnet = (49.0 kg) * (0.800 m/s^2) = 39.2 N
Now, let's consider the tension in the rope (T). The tension in the rope is acting along the ramp and can be calculated using the formula T = Fnet + Ff.
T = (39.2 N) + (135.5 N) = 174.7 N
Therefore, the tension in the rope is approximately 174.7 N.
To find the tension in the rope, we can start by analyzing the forces acting on the sofa. There are four forces acting on the sofa: the force of gravity (mg), the normal force (N) exerted by the ramp, the tension in the rope (T), and the kinetic friction force (fk).
Let's break down the forces acting on the sofa:
1. Force of gravity (mg): This force acts vertically downwards and can be calculated using the equation: mg = (mass of the sofa) * (acceleration due to gravity).
2. Normal force (N): This force acts perpendicular to the surface of the ramp. Since the sofa is on a ramp inclined at an angle of 30.0° to the horizontal, the normal force can be calculated using the equation: N = (mass of the sofa) * (acceleration due to gravity) * cos(30.0°).
3. Tension in the rope (T): This force acts parallel to the surface of the ramp. We are asked to find this force.
4. Kinetic friction force (fk): This force acts opposite to the direction of motion and can be calculated using the equation: fk = (coefficient of kinetic friction) * N.
According to Newton's second law, the net force acting on the sofa is equal to the mass of the sofa multiplied by its acceleration. This can be expressed as:
Net force = mass * acceleration
Taking the forces into account:
T - fk = mass * acceleration
Now, let's substitute the values we know into the equation:
T - (coefficient of kinetic friction) * N = mass * acceleration
mass = 49.0 kg (given)
acceleration = 0.800 m/s^2 (given)
coefficient of kinetic friction = 0.320 (given)
Substituting the values, we have:
T - (0.320) * [(mass) * (acceleration due to gravity) * cos(30.0°)] = (mass) * (acceleration)
Next, we need to calculate the values of (mass * acceleration due to gravity) and (cos(30.0°)):
(mass * acceleration due to gravity) = (49.0 kg) * (9.8 m/s^2) ≈ 480.2 N
cos(30.0°) ≈ 0.866
Now, substitute the values into the equation:
T - (0.320) * (480.2 N) * (0.866) = (49.0 kg) * (0.800 m/s^2)
T - 132.224 N = 39.2 N
Finally, solve for T by adding 132.224 N to both sides of the equation:
T = 39.2 N + 132.224 N
T ≈ 171.4 N
Therefore, the tension in the rope is approximately 171.4 N.