A light string is attached to both a hanging mass m1 = 5 kg and to a pulley of mass m2= 6 kg, and radius R = 0.75 m. If the pulley is a solid disc rotating about its center with a moment of inertia of I = 0.5*m2R2, and the string does not slip on the pulley,
(a) what is the acceleration of mass 1? Assume g=10 m/s2
torque=m(g-a)*R=momentinertia*angular acceleratin
but angular acceleration= ma*(2PI)/R
solve for a.
check my thinking
To find the acceleration of mass 1, we can use Newton's second law of motion. Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force on mass 1 is the tension in the string (T) minus the weight of the mass (mg1), where g is the acceleration due to gravity. The tension in the string is also equal to the force of friction between the string and the pulley.
To find the force of friction, we need to calculate the angular acceleration of the pulley.
The torque (τ) exerted on the pulley is equal to the product of the force of friction (f) and the radius of the pulley (R). The torque is also equal to the product of the moment of inertia (I) and the angular acceleration (α). Therefore, we can write the equation as:
τ = f * R = I * α
Plugging in the given values: I = 0.5 * m2 * R^2 and τ = f * R, we can rewrite the equation as:
f * R = 0.5 * m2 * R^2 * α
Simplifying the equation, we get:
f = 0.5 * m2 * R * α
Now, we can use Newton's second law of motion to relate the force of friction to the acceleration of mass 1. The net force on mass 1 is:
T - m1 * g = m1 * a
Where T is the tension in the string, m1 is the mass of mass 1, g is the acceleration due to gravity, and a is the acceleration of mass 1.
Because the string does not slip on the pulley, the tension in the string (T) is equal to the force of friction (f). So we can replace T with f in the equation:
f - m1 * g = m1 * a
Substituting the value of f from the earlier equation:
0.5 * m2 * R * α - m1 * g = m1 * a
Now, we can solve this equation to find the acceleration of mass 1 (a).
The angular acceleration (α) can be related to the linear acceleration (a) using the equation:
α = a / R
Substituting this relation into the equation above:
0.5 * m2 * R * (a / R) - m1 * g = m1 * a
Simplifying the equation:
0.5 * m2 * a - m1 * g = m1 * a
Rearranging the terms:
m2 * a = (m1 + 0.5 * m2) * g
Finally, to find the acceleration of mass 1 (a), we divide both sides of the equation by m2:
a = [(m1 + 0.5 * m2) * g] / m2
Now, we can plug in the given values: m1 = 5 kg, m2 = 6 kg, and g = 10 m/s^2 to calculate the acceleration (a).