please show full steps Problem 1: A ball of mass m = 0.5 kg hangs from the ceili

Question

please show full steps

Problem 1: A ball of mass m = 0.5 kg hangs from the ceiling of a bus. The bus is accelerating along level ground. The string supporting the ball makes a constant angle of ? = 10? with the vertical. Find the tension in the string (force that the string exerts on the ball) and the acceleration of the bus.

Problem 2:

Consider the figure below. Assume that the ring is massless. Rope 1 makes an angle of 40 degree with the vertical, rope 2 makes an angle of 20 degree with the vertical and rope 3 is vertical. Find the tensions in the three massless ropes supporting the sign of mass M = 10kg.

Explanation / Answer

(Problem 2) (Problem 1 is below)

Tension in ropes 1.2,3 respectively T1,T2& T3

Now

T3 = Mg => T3 = 10*9.81 = 98.1 N

Across ring

In vertical direction

T1*cos(40) + T2*cos(20) = T3

=> T1*0.766 + T2*0.94 = 98.1 -------- (i)

and In horizontal direction

=> T1*sin(40) = T2*sin(20)

=> 0.643 *T1 = 0.342*T2

=> T1 = 0.532*T2

Substituting this in (i)

=> T2*(0.94 + 0.532*0.766) = 98.1

=> T2 = 72.8 N

And T1 = 0.532*T2 = 38.73 N

(Problem 1)

so pseudo force = ma in opposite direction

m = 0.5 kg

a = accln of bus

so for horizontal equilibirium

=> T*sin(10) = 0.5*a

and T*cos(10) = 0.5*g => T = 4.98 N

So accln > a = T*sin(10)/0.5 = 1.73 m/s^2

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