A 0.78 kg rock is whirled around in a flat circle at the end of a string of length 64 cm and a radial acceleration of 3.70 m/s^2 (a) What is the linear velocity of string? (b) what is the centripetal force acting on the rock?
Vm= sqrt( grFs )
= sqrt( 9.8 m/s)(47.5m) (0.740)
?
Ac = v^2/R = 3.70
v^2 = 3.7 * 0.64
solve for v
F = m Ac = 0.78 * .70
To solve this problem, we can use the formula for linear velocity in circular motion:
v = rω
where v is the linear velocity, r is the radius of the circle, and ω is the angular velocity. We can also relate angular velocity to radial acceleration using the formula:
a = rω^2
Given that the radial acceleration a is 3.70 m/s^2 and the length of the string is 64 cm (or 0.64 m), we can rearrange the formulas to solve for ω and then substitute it back to find the linear velocity.
(a) Calculating the angular velocity (ω):
a = rω^2
3.70 m/s^2 = 0.64 m * ω^2
ω^2 = 3.70 m/s^2 / 0.64 m
ω^2 ≈ 5.78125 s^-2
ω ≈ √5.78125 s^-2
ω ≈ 2.404 s^-1
(b) Calculating the linear velocity (v):
v = rω
v = 0.64 m * 2.404 s^-1
v ≈ 1.54 m/s
Thus, the linear velocity of the string is approximately 1.54 m/s.
To find the centripetal force acting on the rock, we can use the formula:
F = m * a
where F is the force, m is the mass of the rock, and a is the radial acceleration.
(c) Calculating the centripetal force (F):
F = 0.78 kg * 3.70 m/s^2
F ≈ 2.886 N
So, the centripetal force acting on the rock is approximately 2.886 N.