Two objects are connected by a light string that passes over a frictionless pulley as in the fi�gure. One object lies on a frictionless, smooth incline. In the �figure, m1 = 8.48 kg, m2 = 4.84 kg, and � theta = 40.2 degrees �. When we let go of the mass m1, it accelerates downward, pulling the mass m2 up the incline. Find (a) the magnitude of the acceleration of the objects and (b) the tension in the string.
write the net force equation, clockwise is positive.
Net force= total mass* a
M1*g-M2*g*sinTheta=(M1+M2)a
find a.
Tension: just above M2
m1(g-a)=T
To find the magnitude of the acceleration of the objects, we can start by analyzing the forces acting on them. Let's consider the two objects separately:
For object m1 (the one on the incline):
1. The weight of m1 (mg1) acts vertically downward and can be split into two components:
- The component parallel to the incline, mg1sin(theta), which acts downhill.
- The component perpendicular to the incline, mg1cos(theta), which does not contribute to the motion.
2. The tension in the string (T) acts uphill and opposes the component of the weight acting downhill, mg1sin(theta).
For object m2 (the one hanging freely):
1. The weight of m2 (mg2) acts vertically downward.
2. The tension in the string (T) acts upward.
Since the masses are connected by a light string, the tension in the string will have the same magnitude for both.
(a) To find the magnitude of the acceleration of the objects, we can use Newton's Second Law for each object:
For m1: m1a = mg1sin(theta) - T
For m2: m2a = T - mg2
Since the objects are connected by the string, the magnitudes of their accelerations will be the same.
We can now solve these two equations simultaneously. Rearranging:
For m1: a = (mg1sin(theta) - T) / m1
For m2: a = (T - mg2) / m2
Since a is the same for both objects, we can equate the two expressions:
(mg1sin(theta) - T) / m1 = (T - mg2) / m2
Now, let's substitute the given values:
m1 = 8.48 kg
m2 = 4.84 kg
theta = 40.2 degrees
We also know that acceleration due to gravity is approximately 9.8 m/s^2.
Plugging in the values, we have:
(8.48 * 9.8 * sin(40.2) - T) / 8.48 = (T - 4.84 * 9.8) / 4.84
Simplifying and rearranging the equation, we get:
(T + 4.84T) / 4.84 = (8.48 * 9.8 * sin(40.2) - 4.84 * 9.8) / 8.48
9.84T = (8.48 * 9.8 * sin(40.2) - 4.84 * 9.8) * (4.84 / 8.48)
Solving this equation will give us the tension in the string (T).
(b) To find the tension in the string, we can substitute the calculated value of T back into one of the earlier equations. Let's use the equation for m1:
a = (mg1sin(theta) - T) / m1
Substituting the known values and the calculated value of T, we can solve for a, which will give us the magnitude of the acceleration.
To find the magnitude of the acceleration of the objects and the tension in the string, we can apply the principles of Newtonian physics.
(a) The acceleration of the objects can be determined by analyzing the forces acting on the system. The gravitational force acting on m1 can be resolved into two components: one along the incline (m1 * g * sin(theta)) and one perpendicular to the incline (m1 * g * cos(theta)). The tension in the string exerts a force on m1 in the upward direction, opposing the downward gravitational force. Therefore, the net force acting on m1 is (m1 * g * sin(theta)) - T.
On the other hand, the gravitational force acting on m2 is m2 * g * sin(theta). The tension in the string exerts a force on m2 in the downward direction, opposing the upward gravitational force. Therefore, the net force acting on m2 is T - (m2 * g * sin(theta)).
Using Newton's second law (F = ma), we can set up the following equations:
For m1:
(m1 * g * sin(theta)) - T = m1 * a
For m2:
T - (m2 * g * sin(theta)) = m2 * a
We can now solve these two equations simultaneously to find the acceleration.
(b) To find the tension in the string, we can substitute the value of acceleration (obtained in part a) into one of the above equations and solve for T.
Now, let's substitute the given values into the equations and calculate the answers.
m1 = 8.48 kg
m2 = 4.84 kg
theta = 40.2 degrees
For m1:
(8.48 * 9.8 * sin(40.2)) - T = 8.48 * a
For m2:
T - (4.84 * 9.8 * sin(40.2)) = 4.84 * a
Solving these equations will provide us with the magnitude of acceleration (a) and the tension in the string (T).