Question

An object of mass m1 hangs from a string that passes over a very light fixed pulley P1 as shown in the figure below. The string connects to a second very light pulley P2. A second string passes around this pulley with one end attached to a wall and the other to an object of mass m2 on a frictionless, horizontal table.(a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? (Use any variable or symbol stated above as necessary.) a2=2a1 (b) Find expressions for the tensions in the strings in terms of the masses m1 and m2, and g. T1= T2= (c) Find expressions for the accelerations a1 and a2 in terms of the masses m1 and m2, and g. a1= a2=

Answer 1

a)

here

Now this is a slightly subtle point, but the forces on the pulley must add to zero because the pulley is assumed to be massless . (A net force on it would give it an infinite acceleration!) This condition gives us:

T1 - 2 * T2 = 0........................(1)

newtons second law gives us

m1 * g - T1 = m1 * a1.....................(2)

then we get

a2 = 2 * a1 ........................(3)

b)

T2 = m2 * a2

a2 = 2 * a1

form these equations we get

T2 = 2 * m2 * a2

then putting this into equation (1) we get

T1 - 2 * T2 = T1 - 4 * m2 * a1 = 0

T1 = 4 * m2 * a1

then by using this in the equation 2

m1 * g - 4 * m2 * a1 = m1 * a1

at which point we can solve for a1

m1 *g = a1( 4 * m2 + m1)

a1 = m1 * g / (4*m2 + m1)

from the equation 3 we get

a2 = 2 *a1 = 2*m1 * g / (4*m2 + m1)

then

T2 = m2 * a2

T2 = 2*m1 * m2 * g / ( 4 * m2 + m1)

form the (1) equation T1 = 2 * T2

T1 = 4 * m1 *m2 *g / (4*m2 + m1)

C)

a1 = m1 * g / ( 4* m2 + m1)

a2 = 2*m2 * g /(4*m2 + m1)