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paulimerci
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- Homework Statement
- The cart is released from rest and slides from the top of an inclined frictionless plane of height h. Express all algebraic answers in terms of the given quantities and fundamental constants.
a) Determine the speed of the cart when it reaches the bottom of the incline.
b) After sliding down the incline and across the frictionless horizontal surface, the cart collides with a bumper of negligible mass attached to an ideal spring k. Determine the distance x_m the spring is compressed before the cart and bumper come to rest.
- Relevant Equations
- Conservation of energy
E_i = E_f
I'm using conservation of energy to solve for both a) and b).
a) Initially, the cart is released from rest, which has maximum GPE at the top of the inclined plane. U_g is converted to K.E as it reaches the bottom of the incline.
$$ E_i = E_f $$
$$ U_g = K.E_f$$
$$ 2mgh = \frac {1}{2}2mv^2$$
$$ v = \sqrt {2gh}$$
b) After sliding down the incline, the cart collides with the bumper, which transforms K.E in to spring P.E.
$$ E_i = E_f$$
$$\frac{1}{2}2mv^2 = \frac {1}{2}kx_m^2$$
$$ x_m = \sqrt \frac {2m}{k} v$$
where ##x_m## is the compressed spring distance.
In part a), I don't know how to find velocity in terms of mass because mass gets canceled out in the equation.
a) Initially, the cart is released from rest, which has maximum GPE at the top of the inclined plane. U_g is converted to K.E as it reaches the bottom of the incline.
$$ E_i = E_f $$
$$ U_g = K.E_f$$
$$ 2mgh = \frac {1}{2}2mv^2$$
$$ v = \sqrt {2gh}$$
b) After sliding down the incline, the cart collides with the bumper, which transforms K.E in to spring P.E.
$$ E_i = E_f$$
$$\frac{1}{2}2mv^2 = \frac {1}{2}kx_m^2$$
$$ x_m = \sqrt \frac {2m}{k} v$$
where ##x_m## is the compressed spring distance.
In part a), I don't know how to find velocity in terms of mass because mass gets canceled out in the equation.
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you only need the mass-distrubution "form factor", for a solid cylinder it is 1/2