a mass of 0.2 kg is whirled at 12 revolutions in 2 sec in a horizontal circle at the end of o string0.8m long.(a)angular velocity (b)linear velocity (c) central acceleration (d) central force
angular velocity= angle/time=12*2PI/2 rad/sec
linear velocity= r*angularvelocity
central acceleration= w^2 * r
force= mass* centralacceleration
To find the answers to these questions, we will need to use some basic concepts from rotational motion. Let's break down each question and explain how to find the answers step by step:
(a) Angular velocity (ω) is defined as the rate of change of angular displacement. It is denoted in units of radians per second (rad/s). To find the angular velocity, we need to determine the angle traversed by the mass in a given time.
In this case, the mass undergoes 12 revolutions in 2 seconds. Since one revolution equals 2π radians, we can calculate the total angle traversed as follows:
Total angle = 12 rev x 2π rad/rev = 24π rad
Finally, divide this angle by the time taken (2 seconds) to get the angular velocity:
Angular velocity (ω) = Total angle / Time = 24π rad / 2 s = 12π rad/s
So the angular velocity is 12π rad/s.
(b) Linear velocity (v) is related to angular velocity (ω) and the radius (r) of the circular path by the formula v = ω * r. Linear velocity represents the speed at which the mass is moving along its circular path.
In this case, the length of the string is given as 0.8 meters, which represents the radius (r) of the circular path.
Using the angular velocity (ω) calculated in part (a) and the radius (r), we can find the linear velocity:
Linear velocity (v) = ω * r = (12π rad/s) * (0.8 m) = 9.6π m/s
So the linear velocity is 9.6π m/s.
(c) Central acceleration (ac) is the acceleration of an object moving in a circular path. It is always directed towards the center of the circle and its magnitude is given by ac = v^2 / r, where v is the linear velocity and r is the radius.
Using the linear velocity (v) calculated in part (b) and the radius (r), we can find the central acceleration:
Central acceleration (ac) = v^2 / r = (9.6π m/s)^2 / (0.8 m) = 115.2π m^2/s^2
So the central acceleration is 115.2π m^2/s^2.
(d) Central force (Fc) is the net force acting on an object moving in a circular path. It is directed towards the center of the circle and is given by Fc = m * ac, where m is the mass of the object and ac is the central acceleration.
In this case, the mass is given as 0.2 kg, and we have previously calculated the central acceleration (ac).
Using these values, we can find the central force:
Central force (Fc) = m * ac = (0.2 kg) * (115.2π m^2/s^2) ≈ 72.16 N (rounded to two decimal places)
So the central force is approximately 72.16 N.
To summarize:
(a) Angular velocity: 12π rad/s
(b) Linear velocity: 9.6π m/s
(c) Central acceleration: 115.2π m^2/s^2
(d) Central force: Approximately 72.16 N