A constant
force of 40N is
applied tangentally
to the rim of a wheel
with 20cm radius.
The wheel has a
moment of inertia
30kg.m2. Find
(a)angular
acceleration
(b)angular speed
after 4seconds from
rest (c)the number
of revolutions made
in that 4 seconds
(d)show that the
work done on the
wheel in this
4seconds=kinetic
energy of th wheel
after 4seconds
torque = moment of inertia * angular acceleration
40 * 0.20 = 30 alpha
alpha = .267 radians/sec^2
omega = alpha * t
omega= .267 * 4 = 1.07 radians/sec
angle = (1/2) alpha *t^2
= .5 *.267 *16 = 2.136 radians
divide by 2 pi = .34 revolutions
torque * angle = work in
= 8 * 8.53 = 17.1 Joules
(1/2)I omega^2 = Ke
= .5 * 30 * 1.07^2 = 17.1 Joules
To solve this problem, we can use the following equations:
(a) Angular acceleration (α) is given by the equation:
α = Στ / I
(b) Angular speed (ω) after a certain time can be calculated using the equation:
ω = ω0 + αt
(c) The number of revolutions made can be found using the equation:
θ = ω0t + 0.5αt²,
where θ is the angle in radians.
(d) To show that the work done on the wheel in 4 seconds is equal to the kinetic energy of the wheel after 4 seconds, we can compare the equations for work and kinetic energy.
Now, let's solve the problem step by step:
(a) Angular acceleration (α) can be calculated using the equation:
α = Στ / I,
where Στ is the net torque applied and I is the moment of inertia.
Given:
Net torque (Στ) = 40 N (tangential force) * 0.20 m (radius) = 8 Nm
Moment of inertia (I) = 30 kg.m²
Therefore,
α = 8 Nm / 30 kg.m²
α = 0.27 rad/s²
(b) Angular speed (ω) after 4 seconds from rest can be calculated using the equation:
ω = ω0 + αt,
where ω0 is the initial angular speed (which is zero in this case) and t is the time.
Given:
ω0 = 0 rad/s (from rest)
t = 4 seconds
Therefore,
ω = 0 + 0.27 rad/s² * 4 seconds
ω = 1.08 rad/s
(c) The number of revolutions made in 4 seconds can be found using the equation:
θ = ω0t + 0.5αt²,
where θ is the angle in radians.
Given:
ω0 = 0 rad/s (from rest)
t = 4 seconds
Therefore,
θ = 0 * 4 + 0.5 * 0.27 rad/s² * (4 seconds)²
θ = 0 + 0.5 * 0.27 rad/s² * 16 seconds²
θ = 2.16 radians
Since 1 revolution = 2π radians,
Number of revolutions = 2.16 / (2π) ≈ 0.34 revolutions
(d) To show that the work done on the wheel in 4 seconds is equal to the kinetic energy of the wheel after 4 seconds, we can compare the equations for work and kinetic energy.
Work done (W) is given by the equation:
W = τθ,
where τ is the torque applied and θ is the angle in radians.
Given:
τ = 40 N (tangential force) * 0.20 m (radius) = 8 Nm
θ = 2.16 radians (from part c)
Therefore,
W = 8 Nm * 2.16 radians
W = 17.28 Joules
Kinetic energy (KE) of the wheel after 4 seconds can be calculated using the equation:
KE = 0.5 I ω²,
where I is the moment of inertia and ω is the angular speed.
Given:
I = 30 kg.m²
ω = 1.08 rad/s (from part b)
Therefore,
KE = 0.5 * 30 kg.m² * (1.08 rad/s)²
KE = 16.56 Joules
The work done on the wheel in 4 seconds is approximately equal to the kinetic energy of the wheel after 4 seconds, as shown by the calculations above.
To solve this problem, we can use some basic principles of rotational motion. Let's go step by step to find each part of the problem.
(a) Angular Acceleration:
The torque applied to the wheel can be calculated using the equation:
Torque = Force * Radius
Given:
Force = 40 N
Radius = 0.2 m
Plugging in the values, we get:
Torque = 40 N * 0.2 m = 8 Nm
The relation between torque and angular acceleration is given by:
Torque = Moment of Inertia * Angular Acceleration
Rearranging the equation, we can find the angular acceleration:
Angular Acceleration = Torque / Moment of Inertia
Given:
Moment of Inertia = 30 kg.m^2
Plugging in the values, we get:
Angular Acceleration = 8 Nm / 30 kg.m^2 ≈ 0.27 rad/s^2
So, the angular acceleration is approximately 0.27 rad/s^2.
(b) Angular Speed after 4 seconds:
The angular speed equation is given by:
Angular Speed = Initial Angular Speed + (Angular Acceleration * Time)
Since the wheel starts from rest (initial angular speed is 0), the equation simplifies to:
Angular Speed = Angular Acceleration * Time
Given:
Angular Acceleration = 0.27 rad/s^2
Time = 4 seconds
Plugging in the values, we get:
Angular Speed = 0.27 rad/s^2 * 4 s = 1.08 rad/s
So, the angular speed after 4 seconds is approximately 1.08 rad/s.
(c) Number of revolutions made in 4 seconds:
To find the number of revolutions, we need to convert the angular speed to revolutions per second and then multiply it by the time.
Given:
Angular Speed = 1.08 rad/s
Time = 4 seconds
The relationship between angular speed and revolutions per second is:
1 revolution = 2π rad
Converting from rad/s to revolutions/s:
Revolutions/s = Angular Speed / (2π rad/revolution)
Plugging in the values, we get:
Revolutions/s = 1.08 rad/s / (2π rad) ≈ 0.171 revolutions/s
Finally, we can calculate the total number of revolutions made in 4 seconds by multiplying the revolutions per second by the time:
Number of revolutions = Revolutions/s * Time = 0.171 revolutions/s * 4 s ≈ 0.684 revolutions
So, the wheel makes approximately 0.684 revolutions in 4 seconds.
(d) Work done on the wheel in 4 seconds = Kinetic energy of the wheel after 4 seconds:
The work done on the wheel is equal to the change in its kinetic energy. The kinetic energy of a rotating object is given by the equation:
Kinetic Energy = (1/2) * Moment of Inertia * Angular Speed^2
Given:
Moment of Inertia = 30 kg.m^2
Angular Speed = 1.08 rad/s
Plugging in the values, we get:
Kinetic Energy = (1/2) * 30 kg.m^2 * (1.08 rad/s)^2
Calculating the value, we find:
Kinetic Energy = 16.56 J
Therefore, the work done on the wheel in 4 seconds is approximately 16.56 Joules, which is equal to the kinetic energy of the wheel after 4 seconds.