A car weighs 1800 kg.
[question] Question. A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel. [/question] [solution] solution: Mass of the car, m = 1800 kg Distance between the front and back axles, d = 1.8 m Distance between the C.G. (centre of gravity) and the back axle = 1.05 m The various forces acting on the car are shown in the following figur...
Show that $\mathbf{a} .(\mathbf{b} \times \mathbf{c})$ is equal in magnitude to the volume
[question] Question. Show that $\mathbf{a} .(\mathbf{b} \times \mathbf{c})$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$. [/question] [solution] solution: A parallelepiped with origin O and sides a, b, and c is shown in the following figure. Volume of the given parallelepiped = abc $\overrightarrow{\mathrm{OC}}=\vec{a}$ $\overrightarrow{\mathrm{OB}}=\vec{b}$ $\overrightarrow{\mathrm{OC}}=\vec{c}$ Let $\hat{\mathb...
Show that the area of the triangle contained between the vectors
[question] Question. Show that the area of the triangle contained between the vectors $\mathbf{a}$ and $\mathbf{b}$ is one half of the magnitude of $\mathbf{a} \times \mathbf{b}$. [/question] [solution] solution: Consider two vectors $\overrightarrow{\mathrm{OK}}=|\vec{a}|$ and $\overrightarrow{\mathrm{OM}}=|\vec{b}|$, inclined at an angle $\theta$, as shown in the following figure. In $\Delta \mathrm{OMN}$, we can write the relation: $\sin \theta=\frac{\mathrm{MN}}{\mathrm{OM}}=\frac{\mathrm{MN...
A child sits stationary at one end of a long trolley moving uniformly
[question] Question. A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system? [/question] [solution] solution: No change The child is running arbitrarily on a trolley moving with velocity v. However, the running of the child will produce no effect on the velocity of the centre of mass of the trolley. This is becau...
Give the location of the centre of mass of a
[question] Question. Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body? [/question] [solution] solution: Geometric centre; No The centre of mass (C.M.) is a point where the mass of a body is supposed to be concentrated. For the given geometric shapes having a uniform mass density, the C.M. lies at their respective geometric centres. The centre of mass of a ...