A crate of mass 106 kg is pulled by a rope on a frictionless surface. The rope makes an angle of 40 0 above the horizontal. If the tension on the string is 1183 N, what is the acceleration of the crate?
1183*cos40=106*acceleration
your teacher is too easy, should have added friction.
Draw a FBD
then,
find component of forces in the x and y direction:
F_x=1183*cos(40) = 906.231 N
F_y=1183*sin(40)-1039.86 = -279.442 N
Now find the resultant force:
F_r = sqrt(F_x^2 + F_y^2)
F_r = sqrt(906.231^2 + (-279.442)^2)
F_r = 948.337 N
a = F_r / m
a = 948.337 / 106 = 8.94657 m/s^2
Don: I don't think the mass leaves nor goes below the surface because of weight and the surface is hard, so the only acceleration can be horizontal, no component downward. If the tension were greater, it could have went upward.
Whoa you're right, I read proble wrong did this as if the mass was on an inclined, but only the force is
To find the acceleration of the crate, we need to analyze the forces acting on it. In this case, we have the tension force from the rope and the gravitational force acting on the crate.
First, let's resolve the tension force into its horizontal and vertical components. The vertical component of the tension force counteracts the gravitational force, while the horizontal component creates acceleration.
The vertical component of the tension force can be calculated using the equation:
T_vertical = T * sin(theta)
Where T is the tension in the rope and theta is the angle the rope makes with the horizontal.
Given T = 1183 N and theta = 40°, we can find:
T_vertical = 1183 * sin(40°) ≈ 758.75 N
Next, we need to calculate the gravitational force acting on the crate. The gravitational force can be calculated using the equation:
F_gravity = m * g
Where m is the mass of the crate and g is the acceleration due to gravity (approximated as 9.8 m/s²).
Given m = 106 kg, we can find:
F_gravity = 106 * 9.8 ≈ 1038.8 N
Since the crate is on a frictionless surface, the only horizontal force acting on it is the horizontal component of the tension force.
Now, using Newton's second law of motion, we can relate the net force to the acceleration of the crate:
Net Force = mass * acceleration
The net horizontal force acting on the crate is equal to the horizontal component of the tension force, which can be calculated as:
T_horizontal = T * cos(theta)
Given T = 1183 N and theta = 40°, we can find:
T_horizontal = 1183 * cos(40°) ≈ 904.43 N
Therefore, the net force acting on the crate is:
Net Force = T_horizontal = 904.43 N
Using Newton's second law, we can now solve for acceleration:
Net Force = mass * acceleration
904.43 N = 106 kg * acceleration
Simplifying the equation, we get:
acceleration = 904.43 N / 106 kg ≈ 8.53 m/s²
Hence, the acceleration of the crate is approximately 8.53 m/s².