An object of mass m1= 203 g is on an inclined surface, that is increasing. The angle of the inclined surface is θ = 40o with the horizontal. The object m1 is connected to a second object of mass m2 = 598 g on the adjacent horizontal surface via a pulley at the base of the incline. Further, an external force of magnitude ІFextІ = 3.3 N is exerted on the object of mass m1 trying to pull the m1 over the surface of the incline. We observe both objects to accelerate. Assuming that the surfaces and the pulley are frictionless, and the pulley and the connecting string are massless, what is the tension in the string connecting the two objects?
Well, the net accelerating force is the external force-force of gravity down the plane.
The force down the plane is M2*g*sinTheta.
Net force=totalmass*a
a=Netforce/total mass
tension= a*m2
To find the tension in the string connecting the two objects, we need to analyze the forces acting on each object and then use Newton's second law of motion.
Let's start by identifying the forces on the first object (m1) on the inclined surface.
1. The gravitational force (mg1) acts vertically downward with a magnitude of m1 * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
2. The normal force (N1) acts perpendicular to the inclined surface and balances the component of the gravitational force perpendicular to the surface. It has a magnitude of N1 = m1 * g * cos(θ), where θ is the angle of the inclined surface with respect to the horizontal.
3. The force exerted by the string (T) pulls the object horizontally up the inclined surface.
4. The external force (Fext) is also acting horizontally on the object, trying to pull it down the inclined surface.
Since the surface and the pulley are frictionless, there are no frictional forces to consider.
Now let's analyze the forces on the second object (m2) on the horizontal surface.
1. The gravitational force (mg2) acts vertically downward with a magnitude of m2 * g.
2. The tension in the string (T) pulls the object horizontally towards the inclined surface.
Since the pulley and the connecting string are assumed to be massless, the tension in the string is the same for both objects.
From Newton's second law of motion (F = ma), we can apply it to both objects independently.
For m1:
Sum of forces on m1 = T - Fext - mg1 * sin(θ) = m1 * a, where a is the acceleration of m1.
For m2:
Sum of forces on m2 = T - mg2 = m2 * a, where a is the acceleration of m2.
Since the acceleration of both objects is the same, we can equate their respective expressions:
m1 * a = m2 * a
Simplifying, we get:
T - Fext - mg1 * sin(θ) = T - mg2
By rearranging the equation, we can solve for T:
T = mg2 + Fext - mg1 * sin(θ)
Now we can substitute the given values:
m1 = 203 g = 0.203 kg
m2 = 598 g = 0.598 kg
θ = 40°
Fext = 3.3 N
g ≈ 9.8 m/s²
Calculating the tension:
T = (0.598 kg * 9.8 m/s²) + 3.3 N - (0.203 kg * 9.8 m/s²) * sin(40°)
T ≈ 5.86 N
Therefore, the tension in the string connecting the two objects is approximately 5.86 N.