1. A 100 lb body is pulled up an inclined plane making an angle of 37 degrees with the horizontal by the rope that passes over the pulley at the top of the plane and is fastened to a 70lb. weight. Neglecting the effect of friction, find the acceleration of the body and the tension in the rope.
Let the tension be T and the acceleration be a.
Write separate free body diagram equations for the two weights
T - 100g*sin37 = 100 a
70 g - T = 70 a
170 a = 9.82 g
a = 0.058 m/s^2
Now solve for T
To find the acceleration of the body and the tension in the rope, we can use Newton's laws of motion.
First, we need to resolve the forces acting on the body along the x and y axes.
Along the x-axis, the only force acting is the component of the weight of the body that is parallel to the inclined plane. This force can be calculated using the formula:
Force_x = mg * sin(theta)
Where m is the mass of the body in pounds, g is the acceleration due to gravity (32.2 ft/s^2), and theta is the angle of the inclined plane.
In this case, m = 100 lb and theta = 37 degrees. Plugging in these values, we get:
Force_x = 100 lb * 32.2 ft/s^2 * sin(37 degrees)
Along the y-axis, we have two forces: the component of the weight of the body that is perpendicular to the inclined plane and the tension in the rope.
The component of the weight perpendicular to the inclined plane can be calculated using the formula:
Force_y = mg * cos(theta)
Plugging in the values, we get:
Force_y = 100 lb * 32.2 ft/s^2 * cos(37 degrees)
Now, since the body is being pulled up the inclined plane, the net force acting in the y-axis direction is equal to the tension in the rope minus the component of the weight perpendicular to the inclined plane. This can be written as:
Net force_y = T - Force_y
Finally, we can use Newton's second law of motion to relate the net force to the acceleration of the body:
Net force_y = mass * acceleration
In this case, the mass of the body is 100 lb. Plugging in the values, we get:
T - (100 lb * 32.2 ft/s^2 * cos(37 degrees)) = 100 lb * acceleration
Now, we have two equations:
Force_x = 100 lb * 32.2 ft/s^2 * sin(37 degrees)
T - (100 lb * 32.2 ft/s^2 * cos(37 degrees)) = 100 lb * acceleration
We can solve these equations simultaneously to find the values of acceleration and tension in the rope.