A mass of 2.6 kg lies on a frictionless table, pulled by another mass of 4.7 kg under the influence of Earth’s gravity.
The acceleration of gravity is 9.8 m/s2 .If the acceleration of the object has a magni- tude of 3.5 m/s2, what is the angle θ between the two forces?
Answer in units of ◦
Yes
To find the angle θ between the two forces, we can use the concept of vectors and the laws of motion. Let's break down the problem and solve it step by step.
1. We have two masses, one of 2.6 kg and the other of 4.7 kg, both under the influence of Earth's gravity. Since the table is frictionless, there are no horizontal forces acting on the masses.
2. The gravitational force acting on each mass can be calculated using the formula F = m * g, where F is the force, m is the mass, and g is the acceleration due to gravity. For the 2.6 kg mass, the gravitational force is F1 = 2.6 kg * 9.8 m/s^2 = 25.48 N. For the 4.7 kg mass, the gravitational force is F2 = 4.7 kg * 9.8 m/s^2 = 46.06 N.
3. The net external force can be calculated using the formula Fnet = m * a, where Fnet is the net external force, m is the mass, and a is the acceleration. We are given that the magnitude of the acceleration is 3.5 m/s^2. Since the 2.6 kg mass is on a frictionless table, the net external force acting on it is F1 = m1 * a = 2.6 kg * 3.5 m/s^2 = 9.1 N.
4. Now, let's consider the forces acting on the 2.6 kg mass. The net external force acting on it is F1 = 9.1 N. In addition to that, there is another force acting on it, which is the gravitational force due to the 4.7 kg mass. This force can be calculated using the formula F = m * a, where F is the force, m is the mass, and a is the acceleration. Since the masses are connected, this force will cause both masses to accelerate with the same magnitude. Therefore, the force acting on the 2.6 kg mass due to the 4.7 kg mass is F2 = 4.7 kg * 3.5 m/s^2 = 16.45 N.
5. We can now find the angle θ between the two forces using trigonometry. The two forces F1 and F2 form a right triangle, where F1 is the adjacent side and F2 is the opposite side. Therefore, we can use the tangent function to find θ using the formula tan(θ) = (F2/F1). Rearranging the equation, we get θ = arctan(F2/F1).
6. Plugging in the values, θ = arctan(16.45 N / 9.1 N) = 59.5° (approximately).
Therefore, the angle θ between the two forces is approximately 59.5°.