An object is released from rest at the top of a ramp. If the ramp is frictionless, does the object’s speed at the bottom of the ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is? ot? frictionless?
Solution 3DQ Let us have a look at the following figure to understand the situation better. When the ramp is frictionless, the downward component of acceleration is gsin x. Let h be the height of the ramp. S be the distance moved by the object down the ramp. Therefore, from the equation of motion v = u + 2aS v = 2gsin x.S …..(1)(since u=0) Now, from the figure, sinx = h/S h = Ssin x Substituting this value in equation 1, v = 2gh v = 2gh Therefore, the object’s speed at the bottom depends only on the height. When there is friction, the net acceleration will be = gsin x cos x , is coefficient of friction 2 2 Therefore, from the equation v = u + 2aS 2 v = 2(gsin x cos x)S (since u is 0) 2 v = 2gh 2Scos x v = gh 2Scos x When friction is present, the object’s speed will depend on height and the coefficient of friction. It does not depend on the shape. It will depend on the nature of the ramp’s surface.