Advertisements
Advertisements
Question
Discuss rolling on an inclined plane and arrive at the expression for acceleration.
Advertisements
Solution
Let us assume a round object of mass m and radius R is rolling down an inclined plane without slipping as shown in the figure. There are two forces acting on the object along the inclined plane. One is the component of gravitational force (mg sin θ) and the other is the static frictional force (f). The other component of gravitation force (mg cos θ) is cancelled by the normal force (N) exerted by the plane. As the motion is happening along the incline, we shall write the equation for motion from the free body diagram (FBP) of the object.
Rolling on an inclined plane
For transnational motion, mg sin θ is the supporting force and f is the opposing force, mg sin θ f = ma
For rotational motion, let us take the torque with respect to the center of the object. Then mg sin 0 cannot cause torque as it passes through it but the frictional force f can set torque of Rf = Iα
By using the relation, a = rα, and moment of inertia I = mK2
we get,
Rf = mK2 `a/R`; f = ma `(K^2/R^2)`
Now equation becomes,
mg sin θ – ma `((K^2)/(R^2))` = ma
mg sin θ = ma + ma `(K^2/R^2)`
a `(1+K^2/R^2)` = g sin θ
After rewriting it for acceleration, we get,
a = `(gsinθ)/((1+K^2/R^2))`
We can also find the expression for the final velocity of the rolling object by using the third equation of motion for the inclined plane.
v2 = u2 + 2as. If the body starts rolling from rest, u = 0. When h is the vertical height of the incline, the length of the incline s is, s = `h/sinθ`
`v^2 = 2(g sintheta)/((1 + K^2/R^2))(h/sin theta) = (2gh)/((1 + K^2/R^2))`
By taking square root,
v = `sqrt((2gh)/((1 + K^2/R^2)))`
The time taken for rolling down the incline could also be written from the first equation of motion as, v = u + at. For the object which starts rolling from rest, u = 0. Then,
t = `v/a`
t = `(sqrt((2gh)/((1 + K^2/R^2)))) (((1 + K^2/R^2))/(g sin theta))`
t = `sqrt((2h(1 + K^2/R^2))/(g sin^2theta))`
The equation suggests that for a given incline, the object with the least value of the radius of gyration K will reach the bottom of the incline first.
APPEARS IN
RELATED QUESTIONS
Derive an expression for kinetic energy, when a rigid body is rolling on a horizontal surface without slipping. Hence find kinetic energy for a solid sphere.
Read each statement below carefully, and state, with reasons, if it is true or false;
For perfect rolling motion, work done against friction is zero.
In rear-wheel drive cars, the engine rotates the rear wheels and the front wheels rotate only because the car moves. If such a car accelerates on a horizontal road the friction
(a) on the rear wheels is in the forward direction
(b) on the front wheels is in the backward direction
(c) on the rear wheels has larger magnitude than the friction on the front wheels
(d) on the car is in the backward direction.
A hollow sphere is released from the top of an inclined plane of inclination θ. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding? (b) Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part (a).
Answer in Brief:
A rigid object is rolling down an inclined plane derive the expression for the acceleration along the track and the speed after falling through a certain vertical distance.
A solid sphere rolls down from top of inclined plane, 7m high, without slipping. Its linear speed at the foot of plane is ______. (g = 10 m/s2)
The least coefficient of friction for an inclined plane inclined at angle α with horizontal in order that a solid cylinder will roll down without slipping is ______.
If x = at + bt2, where x is the distance travelled by the body in kilometers while t is the time in seconds, then the unit of b is ______.
A solid sphere of mass 2 kg is rolling on a frictionless horizontal surface with velocity 6m/s. It collides on the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be ______.
(Force constant of the spring = 36 N/m)
A disc of mass 4 kg rolls on a horizontal surface. If its linear speed is 3 m/ s, what is its total kinetic energy?
