Objects of masses m1 = 4.00kg and m2 = 9.00kg are connected by light frictionless pulley as shown. The object m1 is held at rest on the floor , and m2 rests on a fixed incline of θ equal to 40.00. The objects are released from rest, and m2 slides 1.00m down the incline in 4.00s
To find the acceleration of the objects, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
For m1:
The only force acting on m1 is the force of gravity pulling it downward. Since it is held at rest on the floor, the net force on m1 is zero.
Therefore, the acceleration of m1 is zero.
For m2:
The force of gravity can be resolved into two components: one parallel to the incline and one perpendicular to the incline.
The component of gravity parallel to the incline can be calculated by multiplying the mass of m2 (9.00 kg) by the acceleration due to gravity (9.81 m/s^2) and by the sine of the incline angle (θ).
F_parallel = m2 * g * sin(θ)
The component of gravity perpendicular to the incline is calculated by multiplying the mass of m2 (9.00 kg) by the acceleration due to gravity (9.81 m/s^2) and by the cosine of the incline angle (θ).
F_perpendicular = m2 * g * cos(θ)
The net force acting on m2 parallel to the incline is equal to the component of gravity parallel to the incline minus the force of friction.
F_net = F_parallel - F_friction
Since there is no acceleration perpendicular to the incline, the force of friction is equal to F_perpendicular.
F_friction = F_perpendicular = m2 * g * cos(θ)
Now, we can solve for the net force:
F_net = (m2 * g * sin(θ)) - (m2 * g * cos(θ))
The acceleration of m2 is equal to the net force divided by its mass:
a = F_net / m2
Now, substitute the given values:
m2 = 9.00 kg
θ = 40.00 degrees
g = 9.81 m/s^2
Calculate F_parallel:
F_parallel = 9.00 kg * 9.81 m/s^2 * sin(40.00 degrees) = 56.13 N
Calculate F_perpendicular and F_friction:
F_perpendicular = 9.00 kg * 9.81 m/s^2 * cos(40.00 degrees) = 68.01 N
F_friction = F_perpendicular = 68.01 N
Calculate F_net:
F_net = 56.13 N - 68.01 N = -11.88 N
Now, calculate the acceleration:
a = (-11.88 N) / 9.00 kg = -1.32 m/s^2 (Note: The negative sign indicates that the acceleration is in the opposite direction of motion)
Therefore, the acceleration of m2 is -1.32 m/s^2.
To find the acceleration of the system and the tension in the string, we can use the laws of motion and the principles of dynamics.
First, let's find the acceleration of the system:
1. Start by finding the net force acting on the system. The only forces present are the gravitational force on m2 and the tension force T in the string.
- The gravitational force on m2 can be calculated using the equation F_gravity = m2 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The tension force in the string is the same throughout the string and can be represented as T.
- The net force is the difference between the gravitational force and the tension force in the opposite direction, so we have:
Net force = F_gravity - T
2. According to Newton's second law of motion, the net force is equal to the product of the mass m1 and the acceleration a of the system:
Net force = m1 * a
Equating the two expressions for the net force, we have:
m1 * a = F_gravity - T
Now, let's find the tension in the string:
3. The tension in the string can be calculated using the mass m2, the acceleration a, and the angle of the incline θ.
- The inclined plane creates a normal force N perpendicular to the incline. The gravitational force can be divided into two components: one parallel to the incline (mg*sin(θ)), and the other perpendicular to the incline (mg*cos(θ)).
- The net force acting along the incline is the difference between the gravitational force component parallel to the incline and the friction force (which is negligible since the pulley is frictionless).
Net force along the incline = (m2 * g * sin(θ))
- According to Newton's second law applied in the direction of the incline, this net force is equal to the product of the mass m2 and the acceleration a of the system:
Net force along the incline = m2 * a
Equating the two expressions for the net force along the incline, we have:
m2 * a = (m2 * g * sin(θ))
Now that we have the two equations:
m1 * a = F_gravity - T
m2 * a = (m2 * g * sin(θ))
we can substitute the values provided and solve for the acceleration a and the tension T.